A336564 a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.

1, 1, 2, 3, 4, 0, 6, 7, 8, 2, 10, 2, 12, 4, 6, 15, 16, 3, 18, 8, 10, 8, 22, 6, 24, 10, 26, 14, 28, -12, 30, 31, 18, 14, 22, 17, 36, 16, 22, 20, 40, -12, 42, 26, 27, 20, 46, 14, 48, 17, 30, 32, 52, 12, 38, 34, 34, 26, 58, -18, 60, 28, 43, 63, 46, -12, 66, 44, 42, -4, 70, 45, 72, 34, 41, 50, 58, -12, 78, 44, 80, 38, 82, -14, 62

Offset

1,3

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = n - A308135(n) = n - (sigma(n) - A057723(n)).

a(n) = A336563(n) + A033879(n). [Corrected by Georg Fischer, Dec 13 2022]

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065487 - A013661 + 1 = 0.586357... . - Amiram Eldar, Dec 08 2023

Mathematica

f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 1; a[n_] := n - Times @@ f @@@ (fct = FactorInteger[n]) + Times @@ fc @@@ fct; Array[a, 100] (* Amiram Eldar, Dec 08 2023 *)

Prog

(PARI)
A007947(n) = factorback(factorint(n)[, 1]);
A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };
A308135(n) = (sigma(n)-A057723(n));
A336564(n) = (n - A308135(n));

Crossrefs

Cf. A000203, A033879, A057723, A308135, A336563, A336565, A336566.

Cf. A013661, A065487.

Sequence in context: A181578 A065332 A265515 * A129468 A236846 A236847

Adjacent sequences: A336561 A336562 A336563 * A336565 A336566 A336567

Keyword

sign

Author

Antti Karttunen, Jul 27 2020

Status

approved