A069004 Number of times n^2 + s^2 is prime for positive integers s < n.

0, 1, 1, 1, 2, 2, 1, 3, 1, 4, 2, 1, 4, 3, 3, 3, 4, 3, 4, 6, 2, 4, 5, 3, 7, 6, 4, 4, 4, 4, 7, 6, 5, 6, 8, 5, 6, 7, 3, 9, 5, 5, 8, 8, 7, 9, 6, 7, 10, 8, 6, 9, 10, 5, 8, 8, 6, 10, 11, 8, 11, 10, 6, 9, 15, 5, 10, 11, 4, 11, 13, 6, 12, 10, 12, 11, 9, 8, 11, 19, 10, 15, 9, 8, 19, 11, 8, 11, 14, 15, 13

Offset: 1

Author

T. D. Noe, Apr 02 2002

Keywords

• easy
• nonn

Comments

Conjecture: a(n)>0 for all n>1. - Entries checked by Franklin T. Adams-Watters, May 05 2006

The graph of this sequence inspires the following conjecture: A > a(n)/pi(n) > B, where A and B are constants and pi(n) is the prime counting function (A000720). - T. D. Noe, Feb 26 2007

Stronger conjecture: Let pi(n) be the prime counting function (A000720). Then pi(n) >= a(n) >= pi(n)/5 for n>1, with the following equalities: pi(2)=a(2), pi(10)=a(10) and a(12)=pi(12)/5. - T. D. Noe, Feb 26 2007

Records in a(n) are for n = 1, 2, 5, 8, 10, 20, 25, 35, 40, 49, 59, 65, 80, 115, 125, 130, 158, 200, 250, 265, 310, ... - Thomas Ordowski, Mar 05 2017

Number of primes p = (x^2 + y^2)/2 with 0 < x < y such that x + y = 2n. - Thomas Ordowski, Mar 06 2017

Examples

			a(5)=2 because there are 2 values of s (2 and 4) such that 5^2 + s^2 is a prime number.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A036468, A057368, A281543.

Programs

Mathematica

maxN=100; lst={}; For[n=1, n<=maxN, n++, cnt=0; For[d=1, d?PrimeQ],{n,100}] (* _Harvey P. Dale, Mar 01 2023 *)

PARI

a(n) = sum(s=1, n-1, isprime(n^2+s^2)); \\ Michel Marcus, Jan 15 2017

Formula

a(n) = O(n/log(n)). a(n) <= phi(n), a(n) = phi(n) for n = 2, 6, and 10. a(n) <= phi(2n)/2, a(n) = phi(2n)/2 for n = 2, 3, 5, 6, and 10. - Thomas Ordowski, Mar 01 2017

Extensions

Entries checked by Franklin T. Adams-Watters, May 05 2006