A349379 - OEIS

%I #19 Nov 15 2022 02:37:36

%S 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 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 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 Möbius transform of A057521 (powerful part of n).

%C 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 Antti Karttunen, <a href="/A349379/b349379.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A008683(n/d) * A057521(d).

%F a(n) = Sum_{d|n} A000010(n/d) * A349441(d).

%t 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 (PARI)

%o 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(n) = sumdiv(n,d,moebius(n/d)*A057521(d));

%o (Python)

%o from math import prod

%o from sympy import factorint

%o 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 Cf. A000010, A008683, A057521, A349441.

%Y Cf. also A300717.

%K nonn,mult

%O 1,4

%A _Antti Karttunen_, Nov 18 2021