A380498 -id:A380498 - cp's OEIS frontend

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

Ilya Gutkovskiy, Jan 25 2025

nonn

Eric Weisstein's World of Mathematics, Primorial.

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

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

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}]

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

A380614 Product_{n>=1} (1 + x^n)^a(n) = Sum_{n>=0} prime(n)# * x^n.

Original entry on oeis.org

2, 5, 20, 155, 1860, 24970, 444060, 8583935, 202071920, 5992773714, 186947632200, 7001535728810, 288868991951760, 12455290280871150, 587972068547997856, 31327583556949986095, 1856116108295418943020, 113366872636395467452840, 7619343577986975410930880, 541957669076266404650853414

Offset: 1

Ilya Gutkovskiy, Jan 28 2025

nonn

Inverse Weigh transform of primorial numbers.

Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A168246, A305871, A380498, A380613.

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(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; p(n)-b(n, n-1) end:
seq(a(n), n=1..20);  # Alois P. Heinz, Jan 28 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] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = primorial[n] - b[n, n - 1]; Array[a, 20]

cp's OEIS Frontend

A380497 Euler transform of primorial numbers.

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