A061358 Number of ways of writing n = p+q with p, q primes and p >= q.

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 0, 2, 1, 2, 1, 3, 0, 3, 1, 3, 0, 2, 0, 3, 1, 2, 1, 4, 0, 4, 0, 2, 1, 3, 0, 4, 1, 3, 1, 4, 0, 5, 1, 4, 0, 3, 0, 5, 1, 3, 0, 4, 0, 6, 1, 3, 1, 5, 0, 6, 0, 2, 1, 5, 0, 6, 1, 5, 1, 5, 0, 7, 0, 4, 1, 5, 0, 8, 1, 5, 0, 4, 0, 9, 1, 4, 0, 5, 0, 7, 0, 3, 1, 6, 0, 8, 1, 5, 1

Offset

0,11

Comments

For an odd number n, a(n) = 0 if n-2 is not a prime, otherwise a(n) = 1.

For n > 1, a(2n) is at least 1, according to Goldbach's conjecture.

a(A014092(n)) = 0; a(A014091(n)) > 0; a(A067187(n)) = 1. - Reinhard Zumkeller, Nov 22 2004

Number of partitions of n into two primes.

Number of unordered ways of writing n as the sum of two primes.

a(2*n) = A068307(2*n+2). - Reinhard Zumkeller, Aug 08 2009

4*a(n) is the total number of divisors of all primes p and q such that n = p+q and p >= q. - Wesley Ivan Hurt, Mar 05 2016

Indices where a(n) = 0 correspond to A164376 UNION A025584. - Bill McEachen, Jan 31 2024

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to Goldbach conjecture

Formula

G.f.: Sum_{j>0} Sum_{i=1..j} x^(p(i)+p(j)), where p(k) is the k-th prime. - Emeric Deutsch, Apr 03 2006

A065577(n) = a(10^n).

From Wesley Ivan Hurt, Jan 04 2013: (Start)

a(n) = Sum_{i=1..floor(n/2)} A010051(i) * A010051(n-i).

a(n) = Sum_{i=1..floor(n/2)} floor((A010051(i) + A010051(n-i))/2). (End)

a(n) + A062610(n) + A062602(n) = A004526(n). - R. J. Mathar, Sep 10 2021

a(n) = Sum_{k=floor((n-1)^2/4)+1..floor(n^2/4)} c(A339399(2k-1)) * c(A339399(2k)), where c = A010051. - Wesley Ivan Hurt, Jan 19 2022

Example

a(22) = 3 because 22 can be written as 3+19, 5+17 and 11+11.

Maple

g:=sum(sum(x^(ithprime(i)+ithprime(j)), i=1..j), j=1..30): gser:=series(g, x=0, 110): seq(coeff(gser, x, n), n=0..105); # Emeric Deutsch, Apr 03 2006

Mathematica

a[n_] := Length[Select[n - Prime[Range[PrimePi[n/2]]], PrimeQ]]; Table[a[n], {n, 0, 100}] (* Paul Abbott, Jan 11 2005 *)
With[{nn=110}, CoefficientList[Series[Sum[x^(Prime[i]+Prime[j]), {j, nn}, {i, j}], {x, 0, nn}], x]] (* Harvey P. Dale, Aug 17 2017 *)
Table[Count[IntegerPartitions[n, {2}], _?(AllTrue[#, PrimeQ]&)], {n, 0, 110}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 03 2021 *)

Prog

(PARI) a(n)=my(s); forprime(q=2, n\2, s+=isprime(n-q)); s \\ Charles R Greathouse IV, Mar 21 2013
(Python)
from sympy import primerange, isprime, floor
def a(n):
    s=0
    for q in primerange(2, n//2 + 1): s+=isprime(n - q)
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017
(Magma) [#RestrictedPartitions(n, 2, {p:p in PrimesUpTo(1000)}):n in [0..100] ] // Marius A. Burtea, Jan 19 2019

Crossrefs

a(2n) is A045917.

Cf. A067187, A067188, A067189, A067190, A067191, A063610, A073610, A107318.

Column k=2 of A117278.

Sequence in context: A156542 A307990 A066360 * A025866 A259920 A364334

Adjacent sequences: A061355 A061356 A061357 * A061359 A061360 A061361

Keyword

nonn,easy

Author

Amarnath Murthy, Apr 28 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

Comments edited by Zak Seidov, May 28 2014

Status

approved