cp's OEIS Frontend

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

%I A351572 #13 Feb 25 2022 17:03:30
%S A351572 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,
%T A351572 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,
%U A351572 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
%N A351572 Arithmetic derivative of the largest unitary divisor of n that is a square.
%C A351572 All terms are even: see comments in A235992 and observe that the terms of A350388 are all either odd or multiples of 4.
%H A351572 Antti Karttunen, <a href="/A351572/b351572.txt">Table of n, a(n) for n = 1..65537</a>
%F A351572 a(n) = A003415(A350388(n)).
%t A351572 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 *)
%o A351572 (PARI)
%o A351572 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A351572 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); };
%o A351572 A351572(n) = A003415(A350388(n));
%Y A351572 Cf. A003415, A235992, A350388, A351570, A351573.
%K A351572 nonn
%O A351572 1,4
%A A351572 _Antti Karttunen_, Feb 23 2022