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
Keywords
Examples
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.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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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
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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 *)
Formula
G.f.: Product_{k >= 1} 1/(1 - x^(prime(k)*prime(k+1))). - Robert Israel, Dec 09 2016
Comments