A339625 a(n) is the number of ways to write 6*n = p + q with p a lesser twin prime (A001359) and q a greater twin prime (A006512).

0, 1, 2, 3, 2, 3, 2, 4, 2, 3, 2, 4, 4, 3, 4, 0, 4, 2, 6, 5, 2, 4, 2, 5, 4, 4, 4, 6, 2, 6, 2, 4, 6, 5, 12, 3, 6, 2, 4, 8, 6, 8, 8, 2, 6, 3, 6, 10, 4, 13, 2, 6, 4, 4, 10, 4, 10, 4, 6, 3, 4, 6, 10, 5, 8, 1, 0, 6, 2, 12, 4, 6, 6, 2, 10, 3, 10, 6, 6, 7, 2, 8, 4, 6, 6, 0, 6, 6, 6, 9, 2, 6, 2, 5, 6, 4

Offset: 1

Author

J. M. Bergot and Robert Israel, Dec 10 2020

Keywords

• nonn

Comments

If 6*n = p + q, then also 6*n = (p+2) + (q-2), with p+2 a greater and q-2 a lesser twin prime. Thus a(n) is odd if and only if n/2 is in A002822.

Examples

			a(4)=3 because 6*4 = 24 = 5 + 19 = 11 + 13 = 17 + 7 where (5,7), (11,13) and (17,19) are twin prime pairs.
		

Links

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

Crossrefs

Cf. A001359, A006512, A002822, A339630.

a(n)=0 for n in A243956.

Programs

Maple

N:= 600: # for a(1)..a(floor(N/6)))
P:= select(isprime, {seq(i,i=3..N,2)}):
T1:= sort(convert(P intersect map(`-`,P,2),list)):
T2:= map(`+`,T1,2):
V:= Vector(N):
nT:= nops(T1):
for i from 1 to nT do
  for j from 1 to nT do
    v:= T1[i]+T2[j];
    if v > N then break fi;
    V[v]:= V[v]+1;
od od:
seq(V[6*i],i=1..N/6);