A002100 - OEIS

A002100

a(n) = number of partitions of n into semiprimes (more precisely, number of ways of writing n as a sum of products of 2 distinct primes).
(Formerly M0205 N0076)

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 2, 2, 0, 2, 1, 3, 2, 3, 1, 4, 2, 4, 3, 5, 4, 7, 3, 6, 5, 8, 6, 10, 6, 10, 9, 12, 9, 15, 11, 16, 14, 18, 14, 22, 19, 25, 22, 27, 23, 33, 29, 36, 33, 40, 38, 49, 43, 53, 51, 61, 57, 71, 64, 77, 76, 89, 86, 102, 96, 113, 111, 128, 125

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,20

REFERENCES

L. M. Chawla and S. A. Shad, On a restricted partition function t(n) and its table, J. Natural Sciences and Mathematics, 9 (1969), 217-221. Math. Rev. 41 #6761.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

EXAMPLE

a(20) = 2: 20 = 2*3 + 2*7 = 2*5 + 2*5.

MATHEMATICA

a[n_] := SeriesCoefficient[1/Product[If[SquareFreeQ[k] && PrimeNu[k] == 2, 1 - z^k, 1], {k, 1, n}], {z, 0, n}];

Array[a, 100] (* Jean-François Alcover, Nov 26 2020, after PARI *)

PROG

(PARI) a(n)=polcoeff(1/prod(k=1, n, if(issquarefree(k)*if(omega(k)-2, 0, 1), 1-z^k, 1))+O(z^(n+1)), n)

(Haskell)