A002095 Number of partitions of n into nonprime parts. (Formerly M0271 N0094)

1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 13, 17, 19, 26, 28, 37, 40, 52, 58, 73, 79, 102, 113, 139, 154, 191, 210, 258, 284, 345, 384, 462, 509, 614, 679, 805, 893, 1060, 1171, 1382, 1528, 1792, 1988, 2319, 2560, 2986, 3304, 3823, 4231, 4888, 5399, 6219, 6870

Offset

0,5

Comments

Partial sums of A023895. - Emeric Deutsch, Apr 19 2006

Column k=0 of A222656. - Alois P. Heinz, May 29 2013

References

L. M. Chawla and S. A. Shad, On a trio-set of partition functions and their tables, J. Natural Sciences and Mathematics, 9 (1969), 87-96.

A. Murthy, Some new Smarandache sequences, functions and partitions, Smarandache Notions Journal Vol. 11 N. 1-2-3 Spring 2000 (but beware errors).

Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 2.6.

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

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 1001 terms from T. D. Noe)

Formula

G.f.: Product_{i>0} (1-x^prime(i))/(1-x^i). - Vladeta Jovovic, Jul 31 2004

Example

a(6) = 3 from the partitions 6 = 1+1+1+1+1+1 = 4+1+1.

Maple

g:=product((1-x^ithprime(j))/(1-x^j), j=1..60): gser:=series(g, x=0, 60): seq(coeff(gser, x, n), n=0..55); # Emeric Deutsch, Apr 19 2006

Mathematica

NonPrime[n_Integer] := FixedPoint[n + PrimePi[ # ] &, n + PrimePi[n]]; CoefficientList[ Series[1/Product[1 - x^NonPrime[i], {i, 1, 50}], {x, 0, 50}], x]

Prog

(Haskell)
a002095 = p a018252_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, Jan 15 2012
(PARI) first(n)=my(x='x+O('x^(n+1)), pr=1); forprime(p=2, n+1, pr *= (1-x^p)); pr/prod(i=1, n+1, 1-x^i) \\ Charles R Greathouse IV, Jun 23 2017

Crossrefs

Cf. A000607, A018252, A096258.

Sequence in context: A229950 A050318 A130841 * A029017 A035371 A035577

Adjacent sequences: A002092 A002093 A002094 * A002096 A002097 A002098

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 23 1999

Corrected by Robert G. Wilson v, Feb 11 2002

Status

approved