A380498 Inverse Euler transform of primorial numbers.

2, 3, 20, 150, 1860, 24950, 444060, 8583780, 202071920, 5992771854, 186947632200, 7001535703840, 288868991951760, 12455290280427090, 587972068547997856, 31327583556941402160, 1856116108295418943020, 113366872636395265380920, 7619343577986975410930880, 541957669076266398658079700

Offset

1,1

Links

Table of n, a(n) for n=1..20.

Eric Weisstein's World of Mathematics, Primorial.

Formula

Product_{n>=1} 1 / (1 - x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.

Maple

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

Mathematica

primorial[n_] := Product[Prime[j], {j, 1, n}]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i] + j - 1, j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := primorial[n] - b[n, n - 1]; a /@ Range[20]

Crossrefs

Cf. A002110, A030010, A030011, A112354, A380497.

Sequence in context: A375683 A069323 A356884 * A009721 A013340 A218873

Adjacent sequences: A380495 A380496 A380497 * A380499 A380500 A380501

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 25 2025

Status

approved