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On Highly Factorable Numbers
Journal of Number Theory 72, 7691 (1998) Article No. NT982238
On Highly Factorable Numbers Jun Kyo Kim Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea Communicated by Alan C. Woods Received July 28, 1997; revised March 9, 1998
For a positive integer n, let f (n) be the number of multiplicative partions of n. We say that a nutural number n is highly factorable if f (m)< f (n) for all m, 1 ma j then f (np j p i ) f (n). Using this fact, we prove the conjecture of Canfield, Erdo s, and Pormerance: for each fixed k, if n is a large highly factorable number then there are asymptotically exactly 1k(k+1) of the exponents of n which are equal to k. We also answer the questions posed by Canfield et al.: if n, n$ are consecutive highly factorable numbers, then does it follow n$n 1 and f (n$) f (n) 1? 1998 Academic Press Key Words: Partitions; multiplicative partitions.
1. INTRODUCTION For a positive integer n>1, let f (n) be the number of essentially different ways of writing n as a product of factors greater than 1, where two factorizations of a positive integer are said to be essentially same if they differ only in the order of the factors. For example, f (24)=7, since 24=3 } 8=6 } 4=2 } 12=3 } 2 } 4 =2 } 2 } 6=2 } 2 } 2 } 3. A multi-partite number of order j (or a j-partite number) is an ordered j-tuple of non-negative integers, the components of which are not all zero. For a j-partite number n =(n 1 , n 2 , ..., n j ), let P(n )=( p n11 p n22 } } } p nj j ) where p i denotes i th prime. We abide with the convention P(09 )= f (1)=1. For example, P(4)=5, since 4=3+1=2+2=2+1+1=1+1+1+1 and P(2, 1)=4, since (2, 1)=(2, 0)+(0, 1)=(1, 0)+(1, 0)+(0, 1)=(1, 1)+(1, 0). 76 0022-314X98 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
HIGHLY FACTORABLE NUMBERS
77
Let Pl (n 1 , n 2 , ..., n j ) be the number of partitions of (n 1 , n 2 , ..., n j ) into at most l parts. We say that a nutural number n is highly factorable if f(m)< f (n) for all m, 1 m1 is a highly factorable number, then there is some t 1 such that n= p a11 p a22 } } } p at t ,
a 1 a 2 } } } a t 1
Oppenheim [7] considered the problem of the maximal order of f (n) and claimed that the maximal order was n } L(n) &2+o(1), where L(n) :=exp(log n } log 3 n log 2 n). Here log k n denotes the k-fold iteration of the natural logarithm. Canfield et al. [2] noticed that Oppenheim's estimate has some error and showed that the maximal order is n } L(n) &1+o(1). Some works on partitions of bipartite numbers have dealt with the behavior of the function Pj (x, y) on the line x+ y=2n (see, for example, [4, 6]). In Section 2 we characterize the multisets that attain the maximum value of Pj when the sum of all components of the multipartite number is fixed. Canfield et al. [2] proved that if t s 6t7 and n is a sufficiently large highly factorable number, then p 2s |% n and they conjectured that, for each fixed k, if n is a large highly factorable numbers then there are asymptotically exactly 1k(k+1) of the exponents of n which are equal to k. In Section 3 using result of Section 2, we prove their conjecture. They also mentioned a few problems. Two of their questions were: if n, n$ are consecutive highly factorable numbers, does n$n 1? and does f (n$)f (n) 1? We answer these questions in the affirmative.
2. RESTRICTED PARTITIONS OF MULTIPARTITE NUMBERS In this section we consider a problem related with restricted partitions, that is, partitions in which the number of parts is less than some M and the largest part is less than some L. We prove that for a positive integer n= p a11 } } } p at t , if there is i, j t such that a i >a j then f (np j p i ) f (n). In [4], Kim and Hahn proved the case t=2 for above sentence. We extend the prove to the cases t 2. The basic idea of our proof can be found in [4]. To well understand this section, it maybe helpful to read Kim and Hahn's paper [4]. Let p~(L, M, n) denote the number of partitions of n into M parts, with each part L. The following facts are well known ([1], Chapter 5): (a)
p~(L, M, n)=0, if n>ML;
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JUN KYO KIM
(b)
p~(L, M, LM)=1;
(c)
p~(L, M, n)= p~(M, L, n).
We first introduce some sets and notations used in this section. Let N be the set of all positive integers, V n be the set of all n-partite numbers and P1 be a lexicographic ordering in V n . Define a relation P2 on V n by (m 1 , ..., m n ) P2 (m$1 , ..., m$n ) if and only if and
(m 1 +m 2 , m 3 , ..., m n ) P1 (m$1 +m$2 , m$3 , ..., m$n ) (m 1 , m 2 ) P1 (m$1 , m$2 )
if
m 1 +m 2 =m$1 +m$2 .
Then (V n , P2 ) is a totally ordered set. For each (m, n, k9 ) # V t+1 , we define a function , m, n, k9 : P(m, n, k9 ) P (m+n, k9 ) by , m, n, k9 (( x i , y i , u i ) 1r )= ( x i + y i , u i ) 1r , where r, t # N and
{
}
r
P(m, n, w ) := (x i , y i , u i ) r1 : (x i , y i , u i )=(m, n, w ) and ( x i , y i , u i ) r1 j=1
=
is a non-increasing sequence in ( V t+1 , P2 ) ,
{
}
r
P (l, w ) := (x i , u i ) r1 : (x i , u i )=(l, w ) and (x i + y i , u i ) r1 i=1
=
is a non-increasing sequence in ( V t , P1 ) . One can see that , m, n, u is well defined. Lemma 1.
, m, n, u is surjective.
Proof. Without loss of generality, we may assume m, n # N +. For a T=( t i v i ) ri=1 # P(m+n, u ), let k= min
i.
1 i r i j=0 tj m
Let
(x i , y i )=
{
(t i , 0)
\
k
& : 0 j
(0, t i ),
+
t j , : t j &m , j=0
for
i
for
i=k
for
i>k.
Then ( x i , y i , w i ) 1r # P(m, n, u ) and , m, n, u (( x i , y i , w i ) 1r )=T. K
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HIGHLY FACTORABLE NUMBERS
For a set S, let &S& denote the number of elements in S. For (m, n, u ) # V t and T # P(m+n, u ), let S T (m, n, u )=&, &1 m, n, u (T)&. From Lemma 1, we know S T (m, n, u ) 1 for any T # P(m+n, u ) and p(m, n, u ) =&P(m, n, u )&= V # P(m+n, u ) S V (m, n, u ). A polynomial f (q)=a 0 +a 1 q+ } } } +a n q n is called reciprocal if for each i, a i =a n&i , and f (q) is called unimodal if there exists m such that a 0 a 1 a 2 } } } a m a m+1 a m+2 } } } a n . The following two well-known theorems will be used in the course of our proof. Theorem 2 ([1], p. 47). For all k, l, n # N, the generating function G(k, l; q)= : p~(k, l, n) q n n 0
is a reciprocal, unimodal polynomial in q of degree kl. Theorem 3 ([1], p. 46). Let r(q) and s(q) be reciprocal, unimodal polynomials with non-negative coefficients; then r(q) s(q) is also a reciprocal, unimodal polynomial with nonnegative coefficients. If (m, n) # V 2 and l, k # N with m+n=lk, then S ( l) k1 (m, n)= p~(l, k, m). Lemma 4. Let x, k, r be positive integers. If U=(( (l, v )) k1, ( (t i , w i )) 1r ) # P (x, u ) and (l, v )>1 (t i , w i )>1 0, then x
: S U (m, x&m, u ) q m m=0
\
kl
x&kl
= : p~(l, k, i) q i i=0
+\ : S ( j, x&kl& j, u &kv ) q + , j
T
j=0
where T=( (t i , w i )) 1r . Proof. have
From the comparison of coefficients of both sides of (1), we
kl
S U (m, x&m, u )= : p~(l, k, i) S T (m&i, x&kl&m+i, u &kv ). i=0
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JUN KYO KIM
Let n=x&m. Then, by (2), S U (m, n, u ) =&[(( (x j , y j , v )) k1, ( (x~ a , y~ a , w a )) r1 ) # P(m, n, u ) | x j + y j =l for 1 j k and x~ a + y~ a =t a for 1 a r]& kl
= : &[( (x j , y j , v )) k1 # P(i, kl&i, kv ) | x j + y j =l for 1 j k]& i=0
_&[( (x j , y j , w j )) r1 # P(i, kl&i, u &kv ) | x j + y j =t j for 1 j r]& kl
= : S ( l) k1 (i, kl&i) } S T (m&i, n&kl+i, u &kv ) i=0 kl
= : p~(l, k, i) S T (m&i, n&kl+i, u &kv ).
K
i=0
From Lemma 4 we have the following corollary. and T=(( (l 1 , w 1 )) k11, Corollary 5. Let n, x # N +, (x, u ) # V n k2 kr + ( (l 2 , w 2 )) 1 , ..., ( (l r , w r )) 1 ) # P(x, u ), where k i # N for 1 i r and ( (l i , w i )) 1r is a decreasing sequence of n-partite numbers in (V n , >1 ). Then x
r
: S U (m, x&m, u ) q m = ` m=0
i=1
ki li
\ : p~(l , k , j) q + . j
i
i
j=0
Proposition 6. Let n # N +, (x, u ) # V n and T # P (x, u ). Then x
: S T (m, x&m, u ) q m m=0
is a recipocal, unimodal polynomial in q. Proof. The Proposition follows from Theorem 2, Theorem 3 and Corollary 5. K Proposition 7. Let m, e, j # N and u # V t . Then we have (a)
p j (m+e&1, m&e+1, u )& p j (m+e, m&e, u ) p j&1(e&1, m&e+1, u )& p j&2(u ),
(b)
p j (m+e, m&e+1, u )& p j (m+e+1, m&e, u ) p j&1 (e, m&e, u ).
HIGHLY FACTORABLE NUMBERS
Proof.
81
We define a function g: P (x, u ) N by g(T )=r,
where x # N and T=( t i ) 1r . For a bipartite number (2n&x, x), let h(2n&x, x)=&[( y, z) # V 2 | ( y, z){(2n&x& y, x&z) and y+z=n]&. One can see that &h(2n&x, x)&h(2n&x&1, x+1)& 1. By Lemma 1 and Proposition 6, we have p j (m+e&1, m&e+1, u )& p j (m+e, m&e, u ) =
S T (m+e&1, m&e+1, u )&S T (m+e, m&e, u )
: T # P (2m, u ) g(T ) j
S T (m+e&1, m&e+1, u )
: k
T=( t i ) 1 # P (2m, u ), k j t l =(m, 0, 0, ..., 0) for some l=1, 2
&S T (m+e, m&e, u ) =
p j&1(x, m&x, u )& p j&1( y, m& y, u )
: e&1 x m m+e&1 e y m m+e
& p j&2(u )(h(m+e&1, m&e+1)&h(m+e, m&e)) p j&1(e&1, m&e+1, u )& p j&2(u ). This proves (a). (b) can be proved similarly. K Theorem 8. Then
For r, x, j # N, let H(r, x)=[(x 1 , ..., x r ) # V r | x= ri=1 x i ]. max p j ( y )= p j (z ),
y # H(r, x)
where ( z i ) ri=1 is a sequence in H(r, x) such that |z i &z j | 1 for all positive integers i, l r. Proof. Let y # H(r, x) and | y a & y b | 2 for some a, b. Then we can take y $ # H(r, x) so that | y$a & y$b | 1 and y$j = y j for all j{a, b. Continuing y j"| 1 for all positive integers this process, we can take y " so that | y "& i i, j r. By Proposition 7, we know p j ( y ) p j ( y "). Hence the theorem is proved. K
82
JUN KYO KIM
We may ask whether the ratio p(n, m, u ) , p(n&1, m+1, u ) where m>n 1, is bounded for each multipartite number u or not. The answer is affirmative. For fixed m>n 1, we show in the next section m+1 p(n, m, u ) t p(n&1, m+1, u ) n
|u | ,
as
where |u =( u 1 , u 2 , ..., u t ) | = ti=1 u i . From Lemma 1 and Proposition 7, we have Theorem 9. Let n be an integer with prime representation p a11 p a22 } } } p at t . If there is i, j such that a i >a j then f (np j p i ) f (n).
3. LARGE PRIME FACTORS OF A HIGHLY FACTORABLE NUMBER In this section, to prove the conjecture of Canfield et al., we first need to prove Theorem 15. Then by Theorem 9 in Section 2 and Theorem 15, we extend the following theorem. Theorem 10 ([2], p. 20). There is an =>0 such that if n is a large highly factorable number and (1&=) P(n)
(1&$ n ) log 2 n
_ (1&=) log(m+1)& &2,
where $ n =(log 3 n) 2. We may assume k>l. Let :l =
l s+1 l i=1 s&i+1
p
>
p
and
# k, l =: l
p t+1 p t+2 } } } p t+k&l . p s&l p s&l&1 } } } p s&k+1
From Theorem 16, 2 log n>p t >(log n) 1&$n. Thus from the prime number theorem with error term, we know, for large n, p t+k
(1&=)(k+2)
p s
p t+1 p t+2 } } } p t+k <(4 log n) k
(2.2)
Thus for large n 1 r k, l r k, 0 ( p t p s ) k (1+1- log n) 2k ) ps ps ps =( p t p s ) k+2 (1+1- log n) 2k }
ps p 2t
=(m+1) (k+2)(1&=) (1+1- log n) 2k } (1+1- log n) 2k }
ps p 2t
ps p exp(2k- log n) } s . pt pt
86
JUN KYO KIM
Thus for large n the integer n$=n# k, l p s is smaller than n. We now show f(n$)> f (n), contradicting the choice of n as a highly factorable number. By (2.2) and Lemma 15, for large n, we see that f (n# i, l )>n 34 >(n: l p t+1 p t+2 } } } p t+k ) 14
for all i k.
Let $$>0. Then for large n, by the above inequality and Lemma 2, f (n) f (n: l ) f
p s+1
1
\n: p + (m+1) l
1&$$
f
s&l
p s+1 p t+1 . s&1 p s+1
\n: p l
+
Therefore we have f (n# k, l ) (m+1) (k&l )(1&$$) f (n: l ) (m+1) (k&l )(1&$$) f (n). Let F(n) denote the set of factorization on n. For # F(n) if || denote the number of unequal factors in the factorization then f (n# k, l ) :
( || +1) f (n$) } max[ || +1 : # F(n)]
# F(n)
< f (n$)
\
log n +1 . log 2
+
We choose $$=1&
k(1&= 2 ) k= 2 &l = . k&l k&l
From the choice of k and l
\ + log n \log 2+1+ log n > \log 2+1+ log n > \log 2+1+ log n > \log 2+1+
f (n$)>
&1
f (n# k, l ) &1
(m+1) (k&l )(1&$$) f (n) &1
2
(m+1) (1&= ) k f (n) &1
2
(log n) (1&$n )(1+=) (m+1) (1&= ) 2 f (n) &1
2
(log n) 1+=2 (m+1) (1&= ) 2 f (n)> f (n).
This contradiction proves the theorem. K
87
HIGHLY FACTORABLE NUMBERS
Theorem 18. Let m 1 and 1>=>0. If n is a large highly factorable number and p<(m+l ) &1&= P(n), then p m+1 | n. Proof. Say n is a large highly factorable number with the prime factorization n= p a11 p a22 } } } p at t . Let l=[(1+=2) log(m+1) } log 2 n]
and
k=[log 22 n].
Say for some p s , p s (m+1) &1&= p t , we have a s m and a s&1 m+1. By inductive asumption on m and Theorem 18, we may assume a s =m. Let : l = p t+1 p t+2 p t+l+1
#k =
and
p s p s+1 } } } p s+k&1 . p t+l+1 p t+l } } } p t+l&k+1
&2
From Theorem 15, we know (log n) 1&(log3 n)
\
p s+k
1
+
\
p t
1
+
Thus log(: l # k )<(k&l+1)(log p s+k & p t+l&k+1 )+l log p s+k <(k&l+1)(&1&3=4)(log m+1)+l log( p t 2) f (n), contradicting the choice of n as a highly factorable number. Since by Theorem 9, f (n: l ;) f (n) for ; # B and f (np t+1 )= l | n f (l ), we have f (n: l )>&B& f (n)
t+l
\ l + f (n),
88
JUN KYO KIM
where B=[;= p b11 p b22 } } } p bt t : ; | n and ti=1 b i l]. Hence we have f (n$)= f (n) : l # k >(m+1) &k f (n: l )>(m+1) &k
t+l l
\ + f(n).
Thus Theorem 16 and the above inequality. log f (n$)&log f (n)>l log t&l log l&log(m+1) } log 22 n log 2 n 1+=2
\ + log n log n >l log n& . &4 log n& \ log n 1+=2 +
>l log p t &log 2 p t &log 2&log l&
2
2 2 3
2
3
This contradiction proves the theorem. K By Theorems 17 and 18, we have the following theorem. Theorem 19 (Canfield et al.'s conjecture). For each fixed k if n is large highly factorable numbers then there are asymptotically exactly 1k(k+1) of the exponents of n which are equal to k. Theorem 20. If n, n$ are consecutive highly factorable numbers, then n$n 1 and f (n$) f (n) 1. Proof. First fix k>2 and =>0. Let n be a large highly factorable number with the prime factorization n= p a11 p a22 } } } p at t . Let s= max i, ai k+2
u=min i
n$=np u p s .
and
ai k
By 15, for each large highly factorable number n, we have f(n$)> f (n)
n$>n.
and
By Theorem 19 and Theorem 15, we may assume |n$n| <1+
2 k
89
HIGHLY FACTORABLE NUMBERS
and (k&2) f (n$) f
\
n$
p t+1 p t+1 =f n (k+3) f (n). pu ps
+ \
+
From the definition of a highly factorable number there is a highly factorable number n" such that n
APPENDIX For the convenience of readers, we reproduce the following table of the highly factorable integers less than 10 6 from Canfield et al. [2, Table 1]. TABLE 1
n 1 4 8 12 16 24 36 48 72 96 120 144 192 216 240 288 360 432 480 576 720 960 1080 1152 1440 2160 2880 4320
number of factorizations of n 1 2 3 4 5 7 9 12 16 19 21 29 30 31 38 47 52 57 64 77 98 105 109 118 171 212 289 382
exponents in the prime decomposition of n none 2 3 2 1 4 3 1 2 2 4 1 3 2 5 1 3 1 1 4 2 6 1 3 3 4 1 1 5 2 3 2 1 4 3 5 1 1 6 2 4 2 1 6 1 1 3 3 1 7 2 5 2 1 4 3 1 6 2 1 5 3 1
90
JUN KYO KIM TABLE 1Continued
n
number of factorizations of n
5040 5760 7200 8640 10080 11520 12960 14400 15120 17280 20160 25920 28800 30240 34560 40320 50400 51840 60480 80640 90720 100800 120960 151200 161280 172800 181440 201600 241920 302400 362880 453600 483840 604800 725760 907200
392 467 484 662 719 737 783 843 907 1097 1261 1386 1397 1713 1768 2116 2179 2343 3079 3444 3681 3930 5288 5413 5447 5653 6756 6767 8785 10001 11830 12042 14166 17617 20003 22711
Note. Reproduced from Canfield et al. [2].
ACKNOWLEDGMENT This research was supported in part by KOSEF.
exponents in the prime decomposition of n 4 7 5 6 5 8 5 6 4 7 6 6 7 5 8 7 5 7 6 8 5 6 7 5 9 8 6 7 8 6 7 5 9 7 8 6
2 2 2 3 2 2 4 2 3 3 2 4 2 3 3 2 2 4 3 2 1 2 3 3 2 3 4 2 3 3 4 4 3 3 4 4
1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1
1
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
HIGHLY FACTORABLE NUMBERS
91
REFERENCES 1. G. E. Andrews, ``The Theory of Partitions,'' AddisonWesley, Reading, Massachusetts, 1976. 2. E. R. Canfield, Paul Erdo s, and Carl Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, Journal of Number Theory 17 (1983), 128. 3. V. C. Harris and M. V. Subbarao, On product partitions of integers, Canad. Math. Bull. 34(4) (1991), 474479. 4. J. K. Kim and S. G. Hahn, Partitions of bipartite numbers, Graphs and Combinatorics 13 (1997), 7378. 5. J. K. Kim and S. G. Hahn, Recursive formulae for the multiplicative partitions function, International Journal of Mathematics and Mathematical Sciences, to appear. 6. B. M. Landmam, E. A. Brown, and F. J. Portier, Partitions of bi-partite numbers into at most j parts, Graphs and Combinatorics 8 (1992), 6573. 7. A. Oppenheim, On an arithemetic function, J. London Math. Soc. 114 (1927), 205211. 8. Harold N. Shapiro, ``Introduction to the Theory of Number,'' A WileyInterscience Publication, 1983.
On Highly Factorable Numbers Jun Kyo Kim Department of Mathematics, Korea Advanced Institute of Science and Technology, Taejon 305-701, South Korea Communicated by Alan C. Woods Received July 28, 1997; revised March 9, 1998
For a positive integer n, let f (n) be the number of multiplicative partions of n. We say that a nutural number n is highly factorable if f (m)< f (n) for all m, 1 m
1. INTRODUCTION For a positive integer n>1, let f (n) be the number of essentially different ways of writing n as a product of factors greater than 1, where two factorizations of a positive integer are said to be essentially same if they differ only in the order of the factors. For example, f (24)=7, since 24=3 } 8=6 } 4=2 } 12=3 } 2 } 4 =2 } 2 } 6=2 } 2 } 2 } 3. A multi-partite number of order j (or a j-partite number) is an ordered j-tuple of non-negative integers, the components of which are not all zero. For a j-partite number n =(n 1 , n 2 , ..., n j ), let P(n )=( p n11 p n22 } } } p nj j ) where p i denotes i th prime. We abide with the convention P(09 )= f (1)=1. For example, P(4)=5, since 4=3+1=2+2=2+1+1=1+1+1+1 and P(2, 1)=4, since (2, 1)=(2, 0)+(0, 1)=(1, 0)+(1, 0)+(0, 1)=(1, 1)+(1, 0). 76 0022-314X98 25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved.
HIGHLY FACTORABLE NUMBERS
77
Let Pl (n 1 , n 2 , ..., n j ) be the number of partitions of (n 1 , n 2 , ..., n j ) into at most l parts. We say that a nutural number n is highly factorable if f(m)< f (n) for all m, 1 m
a 1 a 2 } } } a t 1
Oppenheim [7] considered the problem of the maximal order of f (n) and claimed that the maximal order was n } L(n) &2+o(1), where L(n) :=exp(log n } log 3 n log 2 n). Here log k n denotes the k-fold iteration of the natural logarithm. Canfield et al. [2] noticed that Oppenheim's estimate has some error and showed that the maximal order is n } L(n) &1+o(1). Some works on partitions of bipartite numbers have dealt with the behavior of the function Pj (x, y) on the line x+ y=2n (see, for example, [4, 6]). In Section 2 we characterize the multisets that attain the maximum value of Pj when the sum of all components of the multipartite number is fixed. Canfield et al. [2] proved that if t s 6t7 and n is a sufficiently large highly factorable number, then p 2s |% n and they conjectured that, for each fixed k, if n is a large highly factorable numbers then there are asymptotically exactly 1k(k+1) of the exponents of n which are equal to k. In Section 3 using result of Section 2, we prove their conjecture. They also mentioned a few problems. Two of their questions were: if n, n$ are consecutive highly factorable numbers, does n$n 1? and does f (n$)f (n) 1? We answer these questions in the affirmative.
2. RESTRICTED PARTITIONS OF MULTIPARTITE NUMBERS In this section we consider a problem related with restricted partitions, that is, partitions in which the number of parts is less than some M and the largest part is less than some L. We prove that for a positive integer n= p a11 } } } p at t , if there is i, j t such that a i >a j then f (np j p i ) f (n). In [4], Kim and Hahn proved the case t=2 for above sentence. We extend the prove to the cases t 2. The basic idea of our proof can be found in [4]. To well understand this section, it maybe helpful to read Kim and Hahn's paper [4]. Let p~(L, M, n) denote the number of partitions of n into M parts, with each part L. The following facts are well known ([1], Chapter 5): (a)
p~(L, M, n)=0, if n>ML;
78
JUN KYO KIM
(b)
p~(L, M, LM)=1;
(c)
p~(L, M, n)= p~(M, L, n).
We first introduce some sets and notations used in this section. Let N be the set of all positive integers, V n be the set of all n-partite numbers and P1 be a lexicographic ordering in V n . Define a relation P2 on V n by (m 1 , ..., m n ) P2 (m$1 , ..., m$n ) if and only if and
(m 1 +m 2 , m 3 , ..., m n ) P1 (m$1 +m$2 , m$3 , ..., m$n ) (m 1 , m 2 ) P1 (m$1 , m$2 )
if
m 1 +m 2 =m$1 +m$2 .
Then (V n , P2 ) is a totally ordered set. For each (m, n, k9 ) # V t+1 , we define a function , m, n, k9 : P(m, n, k9 ) P (m+n, k9 ) by , m, n, k9 (( x i , y i , u i ) 1r )= ( x i + y i , u i ) 1r , where r, t # N and
{
}
r
P(m, n, w ) := (x i , y i , u i ) r1 : (x i , y i , u i )=(m, n, w ) and ( x i , y i , u i ) r1 j=1
=
is a non-increasing sequence in ( V t+1 , P2 ) ,
{
}
r
P (l, w ) := (x i , u i ) r1 : (x i , u i )=(l, w ) and (x i + y i , u i ) r1 i=1
=
is a non-increasing sequence in ( V t , P1 ) . One can see that , m, n, u is well defined. Lemma 1.
, m, n, u is surjective.
Proof. Without loss of generality, we may assume m, n # N +. For a T=( t i v i ) ri=1 # P(m+n, u ), let k= min
i.
1 i r i j=0 tj m
Let
(x i , y i )=
{
(t i , 0)
\
k
& : 0 j
(0, t i ),
+
t j , : t j &m , j=0
for
i
for
i=k
for
i>k.
Then ( x i , y i , w i ) 1r # P(m, n, u ) and , m, n, u (( x i , y i , w i ) 1r )=T. K
79
HIGHLY FACTORABLE NUMBERS
For a set S, let &S& denote the number of elements in S. For (m, n, u ) # V t and T # P(m+n, u ), let S T (m, n, u )=&, &1 m, n, u (T)&. From Lemma 1, we know S T (m, n, u ) 1 for any T # P(m+n, u ) and p(m, n, u ) =&P(m, n, u )&= V # P(m+n, u ) S V (m, n, u ). A polynomial f (q)=a 0 +a 1 q+ } } } +a n q n is called reciprocal if for each i, a i =a n&i , and f (q) is called unimodal if there exists m such that a 0 a 1 a 2 } } } a m a m+1 a m+2 } } } a n . The following two well-known theorems will be used in the course of our proof. Theorem 2 ([1], p. 47). For all k, l, n # N, the generating function G(k, l; q)= : p~(k, l, n) q n n 0
is a reciprocal, unimodal polynomial in q of degree kl. Theorem 3 ([1], p. 46). Let r(q) and s(q) be reciprocal, unimodal polynomials with non-negative coefficients; then r(q) s(q) is also a reciprocal, unimodal polynomial with nonnegative coefficients. If (m, n) # V 2 and l, k # N with m+n=lk, then S ( l) k1 (m, n)= p~(l, k, m). Lemma 4. Let x, k, r be positive integers. If U=(( (l, v )) k1, ( (t i , w i )) 1r ) # P (x, u ) and (l, v )>1 (t i , w i )>1 0, then x
: S U (m, x&m, u ) q m m=0
\
kl
x&kl
= : p~(l, k, i) q i i=0
+\ : S ( j, x&kl& j, u &kv ) q + , j
T
j=0
where T=( (t i , w i )) 1r . Proof. have
From the comparison of coefficients of both sides of (1), we
kl
S U (m, x&m, u )= : p~(l, k, i) S T (m&i, x&kl&m+i, u &kv ). i=0
80
JUN KYO KIM
Let n=x&m. Then, by (2), S U (m, n, u ) =&[(( (x j , y j , v )) k1, ( (x~ a , y~ a , w a )) r1 ) # P(m, n, u ) | x j + y j =l for 1 j k and x~ a + y~ a =t a for 1 a r]& kl
= : &[( (x j , y j , v )) k1 # P(i, kl&i, kv ) | x j + y j =l for 1 j k]& i=0
_&[( (x j , y j , w j )) r1 # P(i, kl&i, u &kv ) | x j + y j =t j for 1 j r]& kl
= : S ( l) k1 (i, kl&i) } S T (m&i, n&kl+i, u &kv ) i=0 kl
= : p~(l, k, i) S T (m&i, n&kl+i, u &kv ).
K
i=0
From Lemma 4 we have the following corollary. and T=(( (l 1 , w 1 )) k11, Corollary 5. Let n, x # N +, (x, u ) # V n k2 kr + ( (l 2 , w 2 )) 1 , ..., ( (l r , w r )) 1 ) # P(x, u ), where k i # N for 1 i r and ( (l i , w i )) 1r is a decreasing sequence of n-partite numbers in (V n , >1 ). Then x
r
: S U (m, x&m, u ) q m = ` m=0
i=1
ki li
\ : p~(l , k , j) q + . j
i
i
j=0
Proposition 6. Let n # N +, (x, u ) # V n and T # P (x, u ). Then x
: S T (m, x&m, u ) q m m=0
is a recipocal, unimodal polynomial in q. Proof. The Proposition follows from Theorem 2, Theorem 3 and Corollary 5. K Proposition 7. Let m, e, j # N and u # V t . Then we have (a)
p j (m+e&1, m&e+1, u )& p j (m+e, m&e, u ) p j&1(e&1, m&e+1, u )& p j&2(u ),
(b)
p j (m+e, m&e+1, u )& p j (m+e+1, m&e, u ) p j&1 (e, m&e, u ).
HIGHLY FACTORABLE NUMBERS
Proof.
81
We define a function g: P (x, u ) N by g(T )=r,
where x # N and T=( t i ) 1r . For a bipartite number (2n&x, x), let h(2n&x, x)=&[( y, z) # V 2 | ( y, z){(2n&x& y, x&z) and y+z=n]&. One can see that &h(2n&x, x)&h(2n&x&1, x+1)& 1. By Lemma 1 and Proposition 6, we have p j (m+e&1, m&e+1, u )& p j (m+e, m&e, u ) =
S T (m+e&1, m&e+1, u )&S T (m+e, m&e, u )
: T # P (2m, u ) g(T ) j
S T (m+e&1, m&e+1, u )
: k
T=( t i ) 1 # P (2m, u ), k j t l =(m, 0, 0, ..., 0) for some l=1, 2
&S T (m+e, m&e, u ) =
p j&1(x, m&x, u )& p j&1( y, m& y, u )
: e&1 x m m+e&1 e y m m+e
& p j&2(u )(h(m+e&1, m&e+1)&h(m+e, m&e)) p j&1(e&1, m&e+1, u )& p j&2(u ). This proves (a). (b) can be proved similarly. K Theorem 8. Then
For r, x, j # N, let H(r, x)=[(x 1 , ..., x r ) # V r | x= ri=1 x i ]. max p j ( y )= p j (z ),
y # H(r, x)
where ( z i ) ri=1 is a sequence in H(r, x) such that |z i &z j | 1 for all positive integers i, l r. Proof. Let y # H(r, x) and | y a & y b | 2 for some a, b. Then we can take y $ # H(r, x) so that | y$a & y$b | 1 and y$j = y j for all j{a, b. Continuing y j"| 1 for all positive integers this process, we can take y " so that | y "& i i, j r. By Proposition 7, we know p j ( y ) p j ( y "). Hence the theorem is proved. K
82
JUN KYO KIM
We may ask whether the ratio p(n, m, u ) , p(n&1, m+1, u ) where m>n 1, is bounded for each multipartite number u or not. The answer is affirmative. For fixed m>n 1, we show in the next section m+1 p(n, m, u ) t p(n&1, m+1, u ) n
|u | ,
as
where |u =( u 1 , u 2 , ..., u t ) | = ti=1 u i . From Lemma 1 and Proposition 7, we have Theorem 9. Let n be an integer with prime representation p a11 p a22 } } } p at t . If there is i, j such that a i >a j then f (np j p i ) f (n).
3. LARGE PRIME FACTORS OF A HIGHLY FACTORABLE NUMBER In this section, to prove the conjecture of Canfield et al., we first need to prove Theorem 15. Then by Theorem 9 in Section 2 and Theorem 15, we extend the following theorem. Theorem 10 ([2], p. 20). There is an =>0 such that if n is a large highly factorable number and (1&=) P(n)
1 with the convention P(1)=1. Theorem 11 (see [2] and [5], Theorem 4). Let n, t be positive integers and p be a prime number such that p |% m. Then t
_((i, l ) < ) if (i, l ) < l|m
tf (mp t )= : : i=1
m
\ l p +,
where, for positive integers m and n, (m, n) < :=
max k|m n 1k is an integer
k.
t&i
83
HIGHLY FACTORABLE NUMBERS
Lemma 12. Let n, t be positive integers and p, q be prime numbers such that gcd( pq, n)=1. Then f(np t&1q) t } f (np t ) f (np t&1q)+t 2(t&1) f (np t&1 ). Proof.
From Theorem 11 and the fact i&_(k)k 1 for k | i, we have t
\
t } f (np t ) : : f i=1 l | n
n t&i =: f p l d|n
+
\
np t&1 = f (np t&1q). d
+
On the other hand, from Theorem 11 and the fact i } _(k)k _(i) for k | i, and the above inequality, we have t
t } f (np t ) : : _(i) f (lp t&i ) i=1 l | n t
= f (np t&1q)+ : : (_(i)&1) f (lp t&i ) i=2 l | n t&1
f (np t&1q)+t 2 : : f (lp t&1&i ) i=1 l | n
f (np
t&1
q)+t (t&1) f(np t&1 ). 2
K
Theorem 13 ([8], p. 348). Let d(n) be the number of divisors of n. Then lim sup n
Lemma 14. Then
log d(n) log 2 n =log 2. log n
Let = be a positive real number such that 1>=>0 and m 2.
lim
n hold for f (n)>n =
f (ng(n)) , f (n)
where g(n) be the least prime p such that p |% n. Proof. We first show f t(n) d 2t&1(n), where d(n) is the number of divisors of n and f t(n) denotes the number of multiplicative partitions of n which has at most t distinct factors. Notice that f 1(n) d(n). Suppose ft(n) d 2t&1(n). Then f t+1(n) : f 1(l ) f t l |n
n n : d(l ) d 2t&1 d 2t+1(n). l l l|n
\+
\+
84
JUN KYO KIM
Hence f t(n) d 2t&1(n) by induction. Let kn=
1
= log n
\2+2 log d(n) ,
where w:x is the largest integer not exceeding :. Then f kn(n) d 2kn &1(n) n =. Let F(n) be the set of all partitions of n. Since d 2kn &1(n)>2d 2kn &3(n), at least half of the elements of the set F(n) have k n distinct factors. Therefore if || denotes the number of unequal factors in the factorization , then by Theorem 13, we have, for large n, 1 f (np) = : ( || +1) f (n) f (n) # F(n) >
1 k n &[ # F(n) | || k n ]& f (n)
>
1 f (n) k n k = f (n) n 2 2
>
log 2 n 4
as
n .
K
(2.1)
From Lemmas 12 and 14 we have the following theorem. Theorem 15. Let = be a positive real number such that 1>=>0 and t be a positive integer. Then lim
n hold for f (n) n =
f (nr t&1 sn) n =t, t f (nr n )
where r n = g(n), s n = g(nr n ) and g(n) be the least prime p such that p |% n. Proof. Since lim n log r n log n=0, we may assume f (nr tn )>(nr tn ) =2. For large n, from Lemma 12 and (2.1), we have 0 t
t&1 f (nr tn ) 8t 3 ) 3 f (nr n &1 t < . f (nr t&1 sn) f (nr t&1 s n ) = log 2(nr t&1 ) n n n
This proves the theorem. K
85
HIGHLY FACTORABLE NUMBERS
Theorem 16 ([2]). For each large highly factorable number n, P(n) satisfies &2
(log n) 1&(log3 n)
log n <2 log n. log 10 2 n
By Lemma 15, Theorem 10 can be extended as follows. Theorem 17. Let m 1 and =>0. If n is a large highly factorable number and (m+1) &1+= P(n)
(1&$ n ) log 2 n
_ (1&=) log(m+1)& &2,
where $ n =(log 3 n) 2. We may assume k>l. Let :l =
l s+1 l i=1 s&i+1
p
>
p
and
# k, l =: l
p t+1 p t+2 } } } p t+k&l . p s&l p s&l&1 } } } p s&k+1
From Theorem 16, 2 log n>p t >(log n) 1&$n. Thus from the prime number theorem with error term, we know, for large n, p t+k
(1&=)(k+2)
p s
p t+1 p t+2 } } } p t+k <(4 log n) k
(2.2)
Thus for large n 1 r k, l r k, 0 ( p t p s ) k (1+1- log n) 2k ) ps ps ps =( p t p s ) k+2 (1+1- log n) 2k }
ps p 2t
=(m+1) (k+2)(1&=) (1+1- log n) 2k } (1+1- log n) 2k }
ps p 2t
ps p exp(2k- log n) } s . pt pt
86
JUN KYO KIM
Thus for large n the integer n$=n# k, l p s is smaller than n. We now show f(n$)> f (n), contradicting the choice of n as a highly factorable number. By (2.2) and Lemma 15, for large n, we see that f (n# i, l )>n 34 >(n: l p t+1 p t+2 } } } p t+k ) 14
for all i k.
Let $$>0. Then for large n, by the above inequality and Lemma 2, f (n) f (n: l ) f
p s+1
1
\n: p + (m+1) l
1&$$
f
s&l
p s+1 p t+1 . s&1 p s+1
\n: p l
+
Therefore we have f (n# k, l ) (m+1) (k&l )(1&$$) f (n: l ) (m+1) (k&l )(1&$$) f (n). Let F(n) denote the set of factorization on n. For # F(n) if || denote the number of unequal factors in the factorization then f (n# k, l ) :
( || +1) f (n$) } max[ || +1 : # F(n)]
# F(n)
< f (n$)
\
log n +1 . log 2
+
We choose $$=1&
k(1&= 2 ) k= 2 &l = . k&l k&l
From the choice of k and l
\ + log n \log 2+1+ log n > \log 2+1+ log n > \log 2+1+ log n > \log 2+1+
f (n$)>
&1
f (n# k, l ) &1
(m+1) (k&l )(1&$$) f (n) &1
2
(m+1) (1&= ) k f (n) &1
2
(log n) (1&$n )(1+=) (m+1) (1&= ) 2 f (n) &1
2
(log n) 1+=2 (m+1) (1&= ) 2 f (n)> f (n).
This contradiction proves the theorem. K
87
HIGHLY FACTORABLE NUMBERS
Theorem 18. Let m 1 and 1>=>0. If n is a large highly factorable number and p<(m+l ) &1&= P(n), then p m+1 | n. Proof. Say n is a large highly factorable number with the prime factorization n= p a11 p a22 } } } p at t . Let l=[(1+=2) log(m+1) } log 2 n]
and
k=[log 22 n].
Say for some p s , p s (m+1) &1&= p t , we have a s m and a s&1 m+1. By inductive asumption on m and Theorem 18, we may assume a s =m. Let : l = p t+1 p t+2 p t+l+1
#k =
and
p s p s+1 } } } p s+k&1 . p t+l+1 p t+l } } } p t+l&k+1
&2
From Theorem 15, we know (log n) 1&(log3 n)
\
p s+k
1
+
\
p t
1
+
Thus log(: l # k )<(k&l+1)(log p s+k & p t+l&k+1 )+l log p s+k <(k&l+1)(&1&3=4)(log m+1)+l log( p t 2)
t+l
\ l + f (n),
88
JUN KYO KIM
where B=[;= p b11 p b22 } } } p bt t : ; | n and ti=1 b i l]. Hence we have f (n$)= f (n) : l # k >(m+1) &k f (n: l )>(m+1) &k
t+l l
\ + f(n).
Thus Theorem 16 and the above inequality. log f (n$)&log f (n)>l log t&l log l&log(m+1) } log 22 n log 2 n 1+=2
\ + log n log n >l log n& . &4 log n& \ log n 1+=2 +
>l log p t &log 2 p t &log 2&log l&
2
2 2 3
2
3
This contradiction proves the theorem. K By Theorems 17 and 18, we have the following theorem. Theorem 19 (Canfield et al.'s conjecture). For each fixed k if n is large highly factorable numbers then there are asymptotically exactly 1k(k+1) of the exponents of n which are equal to k. Theorem 20. If n, n$ are consecutive highly factorable numbers, then n$n 1 and f (n$) f (n) 1. Proof. First fix k>2 and =>0. Let n be a large highly factorable number with the prime factorization n= p a11 p a22 } } } p at t . Let s= max i, ai k+2
u=min i
n$=np u p s .
and
ai k
By 15, for each large highly factorable number n, we have f(n$)> f (n)
n$>n.
and
By Theorem 19 and Theorem 15, we may assume |n$n| <1+
2 k
89
HIGHLY FACTORABLE NUMBERS
and (k&2) f (n$) f
\
n$
p t+1 p t+1 =f n (k+3) f (n). pu ps
+ \
+
From the definition of a highly factorable number there is a highly factorable number n" such that n
APPENDIX For the convenience of readers, we reproduce the following table of the highly factorable integers less than 10 6 from Canfield et al. [2, Table 1]. TABLE 1
n 1 4 8 12 16 24 36 48 72 96 120 144 192 216 240 288 360 432 480 576 720 960 1080 1152 1440 2160 2880 4320
number of factorizations of n 1 2 3 4 5 7 9 12 16 19 21 29 30 31 38 47 52 57 64 77 98 105 109 118 171 212 289 382
exponents in the prime decomposition of n none 2 3 2 1 4 3 1 2 2 4 1 3 2 5 1 3 1 1 4 2 6 1 3 3 4 1 1 5 2 3 2 1 4 3 5 1 1 6 2 4 2 1 6 1 1 3 3 1 7 2 5 2 1 4 3 1 6 2 1 5 3 1
90
JUN KYO KIM TABLE 1Continued
n
number of factorizations of n
5040 5760 7200 8640 10080 11520 12960 14400 15120 17280 20160 25920 28800 30240 34560 40320 50400 51840 60480 80640 90720 100800 120960 151200 161280 172800 181440 201600 241920 302400 362880 453600 483840 604800 725760 907200
392 467 484 662 719 737 783 843 907 1097 1261 1386 1397 1713 1768 2116 2179 2343 3079 3444 3681 3930 5288 5413 5447 5653 6756 6767 8785 10001 11830 12042 14166 17617 20003 22711
Note. Reproduced from Canfield et al. [2].
ACKNOWLEDGMENT This research was supported in part by KOSEF.
exponents in the prime decomposition of n 4 7 5 6 5 8 5 6 4 7 6 6 7 5 8 7 5 7 6 8 5 6 7 5 9 8 6 7 8 6 7 5 9 7 8 6
2 2 2 3 2 2 4 2 3 3 2 4 2 3 3 2 2 4 3 2 1 2 3 3 2 3 4 2 3 3 4 4 3 3 4 4
1 1 2 1 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2
1
1
1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
HIGHLY FACTORABLE NUMBERS
91
REFERENCES 1. G. E. Andrews, ``The Theory of Partitions,'' AddisonWesley, Reading, Massachusetts, 1976. 2. E. R. Canfield, Paul Erdo s, and Carl Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, Journal of Number Theory 17 (1983), 128. 3. V. C. Harris and M. V. Subbarao, On product partitions of integers, Canad. Math. Bull. 34(4) (1991), 474479. 4. J. K. Kim and S. G. Hahn, Partitions of bipartite numbers, Graphs and Combinatorics 13 (1997), 7378. 5. J. K. Kim and S. G. Hahn, Recursive formulae for the multiplicative partitions function, International Journal of Mathematics and Mathematical Sciences, to appear. 6. B. M. Landmam, E. A. Brown, and F. J. Portier, Partitions of bi-partite numbers into at most j parts, Graphs and Combinatorics 8 (1992), 6573. 7. A. Oppenheim, On an arithemetic function, J. London Math. Soc. 114 (1927), 205211. 8. Harold N. Shapiro, ``Introduction to the Theory of Number,'' A WileyInterscience Publication, 1983.