A283529 The number of partitions of n into simple parts.

1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 9, 9, 12, 12, 15, 15, 18, 18, 22, 22, 26, 26, 30, 30, 35, 35, 40, 40, 45, 45, 52, 52, 59, 59, 66, 66, 75, 75, 84, 84, 93, 93, 104, 104, 115, 115, 126, 126, 139, 139, 152, 152, 165, 165, 180, 180, 195, 195, 210, 210, 228, 228

Offset

0,3

Comments

Number of partitions of n where each part is simple, meaning that each part is in A002110.

Links

Giovanni Resta, Table of n, a(n) for n = 0..10000

J. Wang, Reduced phi-partitions of positive integers, Fib. Quart. 31 (4) (1993) 365-369.

Formula

G.f.: 1/Product_{i>=0} (1-x^A002110(i)).

Example

a(6)=5 counts 1+1+1+1+1+1 = 1+1+1+2 = 1+1+2+2 = 2+2+2 =6.
a(7)=5 counts 1+1+1+1+1+1+1 = 1+1+1+1+1+2 = 1+1+1+2+2 = 1+2+2+2 = 1+6.

Maple

isA002110 := proc(n)
    member(n, [1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070]) ;
end proc:
A283529 := proc(n)
    local a, k, issimp, p ;
    a := 0 ;
    for k in combinat[partition](n) do
        issimp := true ;
        for p in k do
            if not isA002110(p) then
                issimp := false;
                break;
            end if;
        end do:
        if issimp then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:

Mathematica

(* It suffices to compute 3 primorials to get 100 correct terms *)
terms = 100; primorials = FoldList[Times, 1, Prime[Range[3]]]; 1/(Times @@ (1 - x^primorials)) + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, May 19 2018 *)

Crossrefs

Cf. A002110, A064986, A283528.

Sequence in context: A085885 A309685 A309683 * A064986 A349675 A029019

Adjacent sequences: A283526 A283527 A283528 * A283530 A283531 A283532

Keyword

nonn

Author

R. J. Mathar, Mar 10 2017

Extensions

a(0)=1 prepended by Alois P. Heinz, Mar 13 2017

Status

approved