A061358 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.