A351942 Arithmetic derivative of A181819(n), where A181819(n) = Product prime(e(i)) when n = Product prime(i)^e(i).

0, 1, 1, 1, 1, 4, 1, 1, 1, 4, 1, 5, 1, 4, 4, 1, 1, 5, 1, 5, 4, 4, 1, 7, 1, 4, 1, 5, 1, 12, 1, 1, 4, 4, 4, 6, 1, 4, 4, 7, 1, 12, 1, 5, 5, 4, 1, 9, 1, 5, 4, 5, 1, 7, 4, 7, 4, 4, 1, 16, 1, 4, 5, 1, 4, 12, 1, 5, 4, 12, 1, 8, 1, 4, 5, 5, 4, 12, 1, 9, 1, 4, 1, 16, 4, 4, 4, 7, 1, 16, 4, 5, 4, 4, 4, 13, 1, 5, 5, 6, 1, 12

Offset: 1

Author

Antti Karttunen, Apr 02 2022

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A003415, A181819, A351944, A351945.

Coincides with A351943 on the positions given by A130091.

Programs

Mathematica

{0}~Join~Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[Apply[Times, Prime[FactorInteger[#][[All, -1]]]]] &, 101, 2] (* Michael De Vlieger, Apr 03 2022 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A351942(n) = A003415(A181819(n));

Formula

a(n) = A003415(A181819(n)).

a(n) = A351944(n) * A351945(n).