A116357 Number of partitions of n into products of two successive primes (A006094).

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 2, 0, 2, 3, 0, 1, 2, 1, 2, 3, 0, 1, 3, 1, 3, 3, 0, 2, 3, 1, 3, 3, 1, 2, 3, 1, 3, 4, 1, 3, 3, 1, 4, 4, 1, 3, 3, 2, 4, 4, 1, 3, 5

Offset

1,30

Comments

a(A116358(n)) = 0; a(A116359(n)) > 0;

a(n) < A101048(n).

Links

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

Formula

G.f.: Product_{k >= 1} 1/(1 - x^(prime(k)*prime(k+1))). - Robert Israel, Dec 09 2016

Example

a(41) = #{2*3 + 5*7} = 1;
a(42) = #{2*3+2*3+2*3+2*3+2*3+2*3+2*3, 2*3+2*3+3*5+3*5} = 2.

Maple

N:= 200: # to get a(1) to a(N)
Primes:= select(isprime, [2, seq(i, i=3..1+floor(sqrt(N)), 2)]):
G:= mul(1/(1 - x^(Primes[i]*Primes[i+1])), i=1..nops(Primes)-1):
S:= series(G, x, N+1):
seq(coeff(S, x, j), j=1..N); # Robert Israel, Dec 09 2016

Mathematica

m = 105; kmax = PrimePi[Sqrt[m]]; Product[1/(1-x^(Prime[k]*Prime[k+1])), {k, 1, kmax}] + O[x]^(m+1) // CoefficientList[#, x]& // Rest (* Jean-François Alcover, Mar 09 2019, after Robert Israel *)

Crossrefs

Cf. A006094, A101048, A116358, A116359, A116360.

Sequence in context: A324816 A321448 A334158 * A035168 A255647 A119241

Adjacent sequences: A116354 A116355 A116356 * A116358 A116359 A116360

Keyword

nonn

Author

Reinhard Zumkeller, Feb 12 2006

Status

approved