A373480 a(n) = A003415(n) mod A001414(n), where A003415 is the arithmetic derivative, and A001414 is the sum of prime factors with multiplicity.

1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 5, 1, 6, 0, 0, 1, 8, 0, 0, 0, 10, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 5, 1, 3, 6, 0, 1, 2, 0, 9, 0, 5, 1, 4, 0, 1, 0, 0, 1, 8, 1, 0, 12, 0, 0, 13, 1, 9, 0, 3, 1, 0, 1, 0, 3, 11, 0, 17, 1, 7, 0, 0, 1, 12, 0, 0, 0, 4, 1, 6, 0, 15, 0, 0, 0, 12, 1, 13, 7, 0, 1, 3, 1, 12, 11

Offset: 2

Author

Antti Karttunen, Jun 08 2024

Keywords

• nonn
• look

Links

• Antti Karttunen, Table of n, a(n) for n = 2..65537

Crossrefs

Cf. A001414, A003415, A373364, A373481, A373482 (after its initial 1, positions of 0's in this sequence).

Programs

Mathematica

Array[Mod[If[#1 < 2, 0, #1  Total[#2/#1 & @@@ #2]], Total[Times @@@ #2]] & @@ {#, FactorInteger[#]} &, 120, 2] (* Michael De Vlieger, Jun 08 2024 *)

PARI

A001414(n) = ((n=factor(n))[, 1]~*n[, 2]); \\ From A001414.
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A373480(n) = (A003415(n)%A001414(n));