A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.

0, 0, 0, 4, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 32, 0, 6, 0, 4, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 4, 6, 0, 0, 32, 14, 10, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 6, 192, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 10, 4, 0, 0, 0, 32, 108, 0, 0, 4, 0, 0, 0, 0, 0, 6, 0, 4, 0, 0, 0, 0, 0, 14, 6, 140

Offset: 1

Author

Antti Karttunen, Feb 23 2022

Keywords

• nonn

Comments

All terms are even: see comments in A235992 and observe that the terms of A350388 are all either odd or multiples of 4.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A235992, A350388, A351570, A351573.

Programs

Mathematica

f1[p_, e_] := If[EvenQ[e], p^e, 1]; f2[p_, e_] := If[EvenQ[e], e/p, 0]; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]))*(Plus @@ f2 @@@ f); Array[a, 100] (* Amiram Eldar, Feb 23 2022 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
A351572(n) = A003415(A350388(n));

Formula

a(n) = A003415(A350388(n)).