login
A283529
The number of partitions of n into simple parts.
2
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
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
KEYWORD
nonn
AUTHOR
R. J. Mathar, Mar 10 2017
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Mar 13 2017
STATUS
approved