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On a Conjecture of Nicolas–Sárközy about Partitions

✍ Scribed by F. Ben saı̈d

Publisher: Elsevier Science
Year: 2002
Tongue: English
Weight: 152 KB
Volume: 95
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2001.2771

No coin nor oath required. For personal study only.

✦ Synopsis

Let N be the set of positive integers, B ¼ fb 1 5 . . . 5b k g & N, N 2 N, and N5b k . For i ¼ 0 or 1, A ¼ A i ðB; NÞ is the set (introduced by Nicolas, Ruzsa, and Sa´rko¨zy, J. Number Theory 73 (1998), 292-317) such that A \ f1; . . . ; Ng ¼ B and pðA; nÞ iðmod2Þ for n 2 N; n4N, where pðA; nÞ denotes the number of partitions of n with parts in A. Let us denote by ðA; nÞ the sum of the divisors of n belonging to A. In this paper, we prove that ðA; 2nÞ mod 4 is periodic with period q 2 multiple of q period of ðA; nÞ mod 2; we also give the sets B & f1; . . . ; 5g and the values of N; N410, for which q 2 6 ¼ q. Moreover, we show that if AðxÞ is the counting function of A then for A ¼ A 0 ðf1; 2; 3g; 3Þ; lim x!1 AðxÞ=x41=4: # 2002 Elsevier Science (USA)

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