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A336563 Sum of proper divisors of n that are divisible by every prime that divides n.

%I A336563 #22 Dec 07 2023 01:42:54
%S A336563 0,0,0,2,0,0,0,6,3,0,0,6,0,0,0,14,0,6,0,10,0,0,0,18,5,0,12,14,0,0,0,
%T A336563 30,0,0,0,36,0,0,0,30,0,0,0,22,15,0,0,42,7,10,0,26,0,24,0,42,0,0,0,30,
%U A336563 0,0,21,62,0,0,0,34,0,0,0,96,0,0,15,38,0,0,0,70,39,0,0,42,0,0,0,66,0,30,0,46,0,0,0,90
%N A336563 Sum of proper divisors of n that are divisible by every prime that divides n.
%H A336563 Antti Karttunen, <a href="/A336563/b336563.txt">Table of n, a(n) for n = 1..10800</a>
%H A336563 Antti Karttunen, <a href="/A336563/a336563.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A336563 a(n) = A057723(n) - n.
%F A336563 a(n) = A007947(n) * A336567(n) = A007947(n) * A001065(A003557(n)).
%F A336563 a(n) = A336564(n) - A033879(n).
%F A336563 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - 1 = 0.231291... . - _Amiram Eldar_, Dec 07 2023
%t A336563 f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* _Amiram Eldar_, May 06 2023 *)
%o A336563 (PARI)
%o A336563 A007947(n) = factorback(factorint(n)[, 1]);
%o A336563 A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
%o A336563 A336563(n) = (A057723(n)-n);
%o A336563 \\ Or just as:
%o A336563 A336563(n) = { my(x=A007947(n),y = n/x); (x*(sigma(y)-y)); };
%Y A336563 Cf. A001065, A003557, A007947, A057723, A033879, A065487, A335341, A336564, A336565, A336566, A336567.
%Y A336563 Cf. A005117 (positions of zeros).
%K A336563 nonn
%O A336563 1,4
%A A336563 _Antti Karttunen_, Jul 27 2020