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

Author

R. J. Mathar, Mar 10 2017

Keywords

• nonn

Comments

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

Examples

			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.
		

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.

Crossrefs

Cf. A002110, A064986, A283528.

Programs

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 *)

Formula

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

Extensions

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