A077608 Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 75, 103, 125, 167, 220, 275, 370, 474, 610, 806, 1028, 1347, 1752, 2253, 2954, 3812, 4944, 6451, 8329, 10841, 14077, 18226, 23720, 30745, 39903, 51857, 67214, 87313, 113340, 147017, 190974

Offset: 0

Author

Philippe Flajolet, Nov 11 2002

Keywords

• nonn

Examples

			a(15) = 8 since 15 = 11+7 = 7+11 = 5+13 = 13+5 = 3+5+7 = 3+7+5 = 5+3+7 = 5+7+3 = 7+3+5 = 7+5+3 and 3,5,7,11 belong to twin pairs.
		

Links

• Iain Fox, Table of n, a(n) for n = 0..8834 (first 1001 terms from Alois P. Heinz)
• P. Flajolet, Publications

Crossrefs

Cf. A001097, A002124, A023360.

Programs

Maple

A077608 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j)* subs([true=1,false=0],evalb(isprime(ithprime(j)-2) or isprime(ithprime(j)+2))),j=2..n+2)),z=0,n+1),z,n): end;

Mathematica

a[n_] := Coefficient[Series[ 1/(1 - Sum[z^Prime[j]*Boole[ PrimeQ[Prime[j] - 2] || PrimeQ[ Prime[j] + 2]], {j, 2, n + 2}]), {z, 0, n + 1}], z, n]; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Nov 09 2012, after Maple *)

PARI

ok(n)={isprime(n) && (isprime(n-2) || isprime(n+2))}
{my(n=60); Vec(1/(1-sum(k=1, n, if(ok(k), x^k, 0))) + O(x*x^n))} \\ Andrew Howroyd, Dec 28 2017

Formula

G.f.: 1/(1 - Sum_{k>=1} x^A001097(k)). - Andrew Howroyd, Dec 28 2017