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A349379 Möbius transform of A057521 (powerful part of n).

%I A349379 #19 Nov 15 2022 02:37:36
%S A349379 1,0,0,3,0,0,0,4,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,24,0,18,0,0,0,0,16,0,
%T A349379 0,0,24,0,0,0,0,0,0,0,0,0,0,0,0,48,0,0,0,0,0,0,0,0,0,0,0,0,0,0,32,0,0,
%U A349379 0,0,0,0,0,32,0,0,0,0,0,0,0,0,54,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,72,0,0
%N A349379 Möbius transform of A057521 (powerful part of n).
%C A349379 Multiplicative with a(p^e) = 0 if e = 1, p^2 - 1 if e = 2 and p^e - p^(e-1) otherwise. - _Amiram Eldar_, Nov 18 2021
%H A349379 Antti Karttunen, <a href="/A349379/b349379.txt">Table of n, a(n) for n = 1..20000</a>
%F A349379 a(n) = Sum_{d|n} A008683(n/d) * A057521(d).
%F A349379 a(n) = Sum_{d|n} A000010(n/d) * A349441(d).
%t A349379 f[p_, e_] := Which[e > 2, p^e - p^(e - 1), e == 2, p^2 - 1, e == 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 18 2021 *)
%o A349379 (PARI)
%o A349379 A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
%o A349379 A349379(n) = sumdiv(n,d,moebius(n/d)*A057521(d));
%o A349379 (Python)
%o A349379 from math import prod
%o A349379 from sympy import factorint
%o A349379 def A349379(n): return prod(0 if e==1 else p**e - (1 if e==2 else p**(e-1)) for p,e in factorint(n).items()) # _Chai Wah Wu_, Nov 14 2022
%Y A349379 Cf. A000010, A008683, A057521, A349441.
%Y A349379 Cf. also A300717.
%K A349379 nonn,mult
%O A349379 1,4
%A A349379 _Antti Karttunen_, Nov 18 2021