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

1,4

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

Links

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

Formula

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

Example

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)

Prog

(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

Crossrefs

Cf. A053185, A063647, A077068.

Sequence in context: A175663 A240672 A352288 * A284975 A219202 A341980

Adjacent sequences: A243013 A243014 A243015 * A243017 A243018 A243019

Keyword

nonn

Author

Derek Orr, May 29 2014

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

Status

approved