A380497 -id:A380497 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jan 25 2025

nonn

Eric Weisstein's World of Mathematics, Primorial.

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

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

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]

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

A380613 Expansion of Product_{k>=1} (1 + x^k)^prime(k)#.

Original entry on oeis.org

1, 2, 7, 42, 291, 2970, 36950, 597100, 11070875, 248103940, 7018494836, 215718595582, 7881561212732, 320881902092122, 13754717161317416, 643588827524430916, 33926485821837232397, 1992916854095359256932, 121393059052727838936847, 8107963745977267426512386, 574571379331620422000295082

Offset: 0

Ilya Gutkovskiy, Jan 28 2025

nonn

Weigh transform of primorial numbers.

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A061152, A261052, A380497, A380614.

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(p(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Jan 28 2025

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

cp's OEIS Frontend

A380498 Inverse Euler transform of primorial numbers.

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