A069360 Number of prime pairs (p,q), p <= q, such that (p+q)/2 = 2*n.

1, 1, 1, 2, 2, 3, 2, 2, 4, 3, 3, 5, 3, 3, 6, 5, 2, 6, 5, 4, 8, 4, 4, 7, 6, 5, 8, 7, 6, 12, 5, 3, 9, 5, 7, 11, 5, 4, 11, 8, 5, 13, 6, 7, 14, 8, 5, 11, 9, 8, 14, 7, 6, 13, 9, 7, 12, 7, 9, 18, 9, 6, 16, 8, 10, 16, 9, 7, 16, 14, 8, 17, 8, 8, 21, 10, 8, 17, 10, 11

Offset

1,4

Comments

The Goldbach conjecture, if true, would imply a(n) > 0.

Row lengths of table A260689, n > 1. - Reinhard Zumkeller, Nov 17 2015

Links

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Goldbach conjecture

Formula

For n > 1: a(n) = #{k | 2*n-k and 2*n+k are prime, 1<=k<=2*n}.

a(n) = Sum_{i=3..2n} isprime(i) * isprime(4n-i) * (sign(4n-i) mod 4), n > 1. - Wesley Ivan Hurt, Dec 18 2016

Example

n=8: there are 16 pairs (i,j) with (i+j)/2=n*2=16; only two of them, (3,29) and (13,19), consist of primes, therefore a(8)=2.

Mathematica

Table[Length[Select[Range[0, 2*n], PrimeQ[2n-#] && PrimeQ[2n+#] &]], {n, 50}] (* Stefan Steinerberger, Nov 30 2007 *)
Table[Boole[n == 1] + Sum[Boole[PrimeQ@ i] Boole[PrimeQ[4 n - i]] Mod[Sign[4 n - i], 4], {i, 3, 2 n}], {n, 80}] (* Michael De Vlieger, Dec 21 2016 *)
Table[Count[IntegerPartitions[4n, {2}], _?(AllTrue[#, PrimeQ]&)], {n, 80}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 09 2018 *)

Prog

(Haskell)
a069360 n = sum [a010051' (4*n-p) | p <- takeWhile (<= 2*n) a000040_list]
-- Reinhard Zumkeller, May 08 2014, Apr 09 2012
(PARI) a(n)=my(s); forprime(p=2, 2*n, s+=isprime(4*n-p)); s \\ Charles R Greathouse IV, Apr 09 2012

Crossrefs

Bisection of A002375.

Cf. A082467 (least k such that n-k and n+k are both primes), A134677 (records), A134678 (where records occur), A135146 (index of first occurrence of n).

Cf. A000040, A010051, A260689.

Sequence in context: A278317 A086454 A339492 * A175509 A213023 A068050

Adjacent sequences: A069357 A069358 A069359 * A069361 A069362 A069363

Keyword

nonn,easy,nice

Author

Reinhard Zumkeller, Apr 15 2002

Extensions

Edited by Klaus Brockhaus, Nov 20 2007

a(1)=1, thanks to Charles R Greathouse IV, who noticed this; b-file adjusted.

Status

approved