A236747 Number of 0 <= k <= sqrt(n) such that n-k and n+k are both prime.

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

Offset: 1

Author

Juri-Stepan Gerasimov, Jan 30 2014

Keywords

• nonn

Comments

Probably a(n) > N for any N and all sufficiently large n. Perhaps a(2591107) is the last 0 in this sequence. - Charles R Greathouse IV, Jan 30 2014

Primes p such that a(p)=1: 2, 3, 7, 11, 13, 17, 19, ... . Juri-Stepan Gerasimov, Feb 02 2014

Examples

			a(3) = 1 because 3 - 0 = 3 and 3 + 0 = 3 are both prime for k = 0;
a(4) = 1 because 4 - 1 = 3 and 4 + 1 = 5 are both prime for k = 1 < sqrt(4) = 2;
a(5) = 2 because 5 - 0 = 5 and 5 + 0 = 5 are both prime for k = 0, 5 - 2 = 3 and 5 + 2 = 7 are both prime for k = 2 < sqrt(5).
		

Links

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

Crossrefs

Cf. A000196, A010051, A061357, A171637.

Programs

Maple

A236767 := proc(n)
    local a,k ;
    a := 0 ;
    for k from 0 to floor(sqrt(n)) do
        if isprime(n-k) and isprime(n+k) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A236767(n),n=1..80) ; # R. J. Mathar, Dec 01 2020

Mathematica

Table[Length[Select[Range[0, Sqrt[n]], PrimeQ[n - #] && PrimeQ[n + #] &]], {n, 100}] (* T. D. Noe, Feb 01 2014 *)

PARI

a(n)=sum(k=0,sqrtint(n),isprime(n-k)&&isprime(n+k)) \\ Charles R Greathouse IV, Jan 30 2014

Scheme

(define (A236747 n) (add (lambda (k) (* (A010051 (- n k)) (A010051 (+ n k)))) 0 (A000196 n)))
;; The following implements sum_{i=lowlim..uplim} intfun(i):
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; From Antti Karttunen, Feb 01 2014

Formula

a(n) = Sum_{k=0..A000196(n)} (A010051(n-k) * A010051(n+k)). - Antti Karttunen, Feb 01 2014

Extensions

Terms recomputed (with corrections) by Antti Karttunen, Feb 01 2014