A342414 a(n) = A003415(n) / gcd(phi(n),A003415(n)), where A003415(n) is the arithmetic derivative of n, and phi is Euler totient function.

0, 1, 1, 2, 1, 5, 1, 3, 1, 7, 1, 4, 1, 3, 1, 4, 1, 7, 1, 3, 5, 13, 1, 11, 1, 5, 3, 8, 1, 31, 1, 5, 7, 19, 1, 5, 1, 7, 2, 17, 1, 41, 1, 12, 13, 25, 1, 7, 1, 9, 5, 7, 1, 9, 2, 23, 11, 31, 1, 23, 1, 11, 17, 6, 3, 61, 1, 9, 13, 59, 1, 13, 1, 13, 11, 20, 3, 71, 1, 11, 2, 43, 1, 31, 11, 15, 4, 7, 1, 41, 5, 24, 17, 49, 1, 17

Offset: 1

Author

Antti Karttunen, Mar 11 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12500
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Crossrefs

Cf. A000010, A003415, A342001, A342008 (positions of ones), A342413, A342415, A342416.

Programs

Mathematica

Array[#1/GCD[#1, #2] & @@ {If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ Abs[#], EulerPhi[#]} &, 96] (* Michael De Vlieger, Mar 11 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342414(n) = { my(u=A003415(n)); (u/gcd(eulerphi(n),u)); };

Formula

a(n) = A003415(n) / A342413(n) = A003415(n) / gcd(A000010(n),A003415(n)).

a(n) = A342001(n) / A342416(n).