cp's OEIS Frontend

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

%I A243016 #17 Jan 20 2025 22:45:12
%S A243016 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,
%T A243016 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,
%U A243016 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
%N A243016 Number of solutions for k*n/(k+n) = p for integer k > 0 and prime p.
%C A243016 It is unknown whether a(6) = 3 is the highest number in this sequence.
%C A243016 No terms higher than 3 among the first 10000 terms. - _Antti Karttunen_, Jan 20 2025
%C A243016 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
%H A243016 Antti Karttunen, <a href="/A243016/b243016.txt">Table of n, a(n) for n = 1..10000</a>
%F A243016 a(n) <= A063647(n). - _Antti Karttunen_, Jan 18 2025
%e A243016 4*k/(4+k) has two solutions: k=4, p=2 and k=12, p=3. Thus a(4) = 2.
%e A243016 From _Antti Karttunen_, Jan 18 2025: (Start)
%e A243016 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.
%e A243016 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)
%o A243016 (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
%Y A243016 Cf. A053185, A063647, A077068.
%K A243016 nonn
%O A243016 1,4
%A A243016 _Derek Orr_, May 29 2014
%E A243016 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