A380497 Euler transform of primorial numbers.

1, 2, 9, 46, 314, 3072, 37641, 603510, 11148030, 249327430, 7040987792, 216220333314, 7895699690498, 321315600822232, 13770543972819903, 644232544408157820, 33954066516677635554, 1994206929690480710244, 121461036181617491970561, 8111955386813996410196454, 574814471423312085719652432

Offset: 0

Author

Ilya Gutkovskiy, Jan 25 2025

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Primorial.

Crossrefs

Cf. A002110, A030009, A030012, A107895, A380498.

Programs

Maple

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      add(d*p(d), d=numtheory[divisors](j)), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 25 2025

Mathematica

nmax = 20; CoefficientList[Series[Product[1/(1 - x^k)^Product[Prime[j], {j, k}], {k, 1, nmax}], {x, 0, nmax}], x]
primorial[n_] := Product[Prime[j], {j, 1, n}]; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d primorial[d], {d, Divisors[j]}] a[n - j], {j, 1, n}]/n]; Table[a[n], {n, 0, 20}]

Formula

G.f.: Product_{k>=1} 1 / (1 - x^k)^prime(k)#.