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A336563
Sum of proper divisors of n that are divisible by every prime that divides n.
12
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, 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, 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
OFFSET
1,4
FORMULA
a(n) = A057723(n) - n.
a(n) = A007947(n) * A336567(n) = A007947(n) * A001065(A003557(n)).
a(n) = A336564(n) - A033879(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - 1 = 0.231291... . - Amiram Eldar, Dec 07 2023
MATHEMATICA
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 *)
PROG
(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A336563(n) = (A057723(n)-n);
\\ Or just as:
A336563(n) = { my(x=A007947(n), y = n/x); (x*(sigma(y)-y)); };
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jul 27 2020
STATUS
approved