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

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..10800

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

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

Cf. A001065, A003557, A007947, A057723, A033879, A065487, A335341, A336564, A336565, A336566, A336567.

Cf. A005117 (positions of zeros).

Sequence in context: A212148 A364988 A358622 * A048146 A372361 A334045

Adjacent sequences: A336560 A336561 A336562 * A336564 A336565 A336566

Keyword

nonn

Author

Antti Karttunen, Jul 27 2020

Status

approved