A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.

0, 0, 1, 2, 0, 3, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Author

Derek Orr, May 29 2014

Keywords

• nonn

Comments

It is unknown whether a(6) = 3 is the highest number in this sequence.

No terms higher than 3 among the first 10000 terms. - Antti Karttunen, Jan 20 2025

a(n) is the number of primes among n-1, n/2 and q, where q satisfies q*(q+1)=n. So a(n) <= 2 for n > 6, and a(n) = 2 iff n != 6 is in A053185 + 1 or A077068. - Jinyuan Wang, Jan 20 2025

Examples

			4*k/(4+k) has two solutions: k=4, p=2 and k=12, p=3. Thus a(4) = 2.
From _Antti Karttunen_, Jan 18 2025: (Start)
For n=3, the ratio (k*n)/(k+n) obtains for k=1..3*(3-1) the values 3/4, 6/5, 3/2, 12/7, 15/8, 2, and only the last one of these is prime, therefore a(3) = 1.
For n=26, the only k such that (k*n)/(k+n) is a prime, is k=26, with (26^2)/(2*26) = 13, therefore a(26) = 1. (End)
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A053185, A063647, A077068.

Programs

PARI

A243016(n) = { my(s); sum(k=1, n*(n-1), s = (k*n)/(k+n); (1==denominator(s) && isprime(s))); }; \\ Edited by Antti Karttunen, Jan 18 2025

Formula

a(n) <= A063647(n). - Antti Karttunen, Jan 18 2025

Extensions

Data section extended up to a(105) and incorrect terms, that were caused by dropping of a(26) and a(27) (first discrepancies at n=26, 28, 30, 34, etc.) corrected by Antti Karttunen, Jan 18 2025