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

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)
a002100 = p a006881_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Mar 21 2014

Crossrefs

Cf. A006881, A073576, A101048.

Sequence in context: A112792 A138319 A217864 * A108352 A346149 A215883

Adjacent sequences: A002097 A002098 A002099 * A002101 A002102 A002103

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Benoit Cloitre, Jun 01 2003

Status

approved