A336564 - OEIS

%I #16 Dec 08 2023 12:31:31

%S 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,

%T 28,-12,30,31,18,14,22,17,36,16,22,20,40,-12,42,26,27,20,46,14,48,17,

%U 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

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

%H Antti Karttunen, <a href="/A336564/b336564.txt">Table of n, a(n) for n = 1..20000</a>

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

%F a(n) = A336563(n) + A033879(n). [Corrected by _Georg Fischer_, Dec 13 2022]

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

%t 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 *)

%o (PARI)

%o A007947(n) = factorback(factorint(n)[, 1]);

%o A057723(n) = { my(r=A007947(n)); (r*sigma(n/r)); };

%o A308135(n) = (sigma(n)-A057723(n));

%o A336564(n) = (n - A308135(n));

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

%Y Cf. A013661, A065487.

%K sign

%O 1,3

%A _Antti Karttunen_, Jul 27 2020