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

1,5

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

Links

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

Formula

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

Example

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).

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 *)

Prog

(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

Crossrefs

Cf. A000196, A010051, A061357, A171637.

Sequence in context: A073490 A279907 A225654 * A194285 A367623 A336676

Adjacent sequences: A236744 A236745 A236746 * A236748 A236749 A236750

Keyword

nonn

Author

Juri-Stepan Gerasimov, Jan 30 2014

Extensions

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

Status

approved