A344987 Number of Goldbach partitions of 2n into 2 primes where the smaller prime has an odd index and the larger prime has an even index.

0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 4, 0, 1, 1, 1, 3, 1, 0, 2, 3, 0, 2, 1, 1, 4, 0, 0, 4, 3, 0, 3, 1, 0, 4, 1, 0, 3, 3, 0, 2, 2, 1, 4, 1, 1, 3, 3, 0, 2, 4, 0, 1, 3, 1, 3, 4, 0, 2, 4, 1, 2, 4

Offset

1,12

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Goldbach Partition

Wikipedia, Goldbach's conjecture

Index entries for sequences related to Goldbach conjecture

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..n} (pi(k) mod 2) * ((pi(2*n-k)+1) mod 2) * c(k) * c(2*n-k), where c(n) is the prime characteristic.

a(n) = A342301(n) - A344986(n).

Example

a(12) = 2; 2*12 = 24 has 2 Goldbach partitions where the smaller prime has an odd index and the larger prime has an even index: (19,5) and (13,11). For example, 19 is the 8th prime and 5 is the 3rd, while 13 is the 6th prime and 11 is the 5th.

Maple

P:= select(isprime, [2, seq(i, i=3..200, 2)]): nP:= nops(P):
V:= Vector(100):
for i from 3 to nP by 2 do
  for j from i+1 to nP by 2 do
    v:= (P[i]+P[j])/2;
    if v > 100 then break fi;
    V[v]:= V[v]+1
od od:
convert(V, list); # Robert Israel, Mar 02 2026

Mathematica

Table[Sum[Mod[PrimePi[k], 2] Mod[PrimePi[2 n - k] + 1, 2] (PrimePi[k] - PrimePi[k - 1]) (PrimePi[2 n - k] - PrimePi[2 n - k - 1]), {k, n}], {n, 100}]

Crossrefs

Cf. A342301, A344986.

Sequence in context: A283669 A220945 A089224 * A359325 A325122 A234578

Adjacent sequences: A344984 A344985 A344986 * A344988 A344989 A344990

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 04 2021

Status

approved