A197132 Euler transform of composite numbers.

1, 4, 16, 52, 157, 434, 1144, 2862, 6906, 16090, 36449, 80430, 173555, 366802, 761102, 1552569, 3118508, 6174461, 12064383, 23283027, 44419855, 83834278, 156626605, 289839251, 531534746, 966483534, 1743164649, 3119864511, 5543030861, 9779552117, 17139055493

Offset: 0

Author

Franklin T. Adams-Watters, Oct 10 2011

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A002808, A030009.

Programs

Maple

N:= 100: # to use composites <= N
comps:= remove(isprime,[$4..N]):
M:= nops(comps):
G:= mul((1-x^k)^(-comps[k]),k=1..M):
S:= series(G, x, M+1):
seq(coeff(S,x,j),j=0..M); # Robert Israel, Jan 30 2018

Mathematica

a[ns_Integer?NonNegative, nf_Integer?NonNegative] := CoefficientList[Series[Product[(1 - x^k)^-FixedPoint[k + PrimePi[#] + 1 &, k], {k, 1, nf}], {x, 0, nf}], x][[ns + 1 ;; nf + 1]]; a[0, 30] (* Robert P. P. McKone, Nov 08 2023 *)

Formula

G.f.: Product_{k>=1} (1-x^k)^-composite(k), where composite(k) = A002808(k) is the k-th composite number.