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A336564
a(n) = n - A308135(n), where A308135(n) is the sum of non-coreful divisors of n.
5
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
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));
KEYWORD
sign
AUTHOR
Antti Karttunen, Jul 27 2020
STATUS
approved