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%I #20 Dec 12 2023 11:27:48
%S 0,1,1,1,1,6,1,1,1,8,1,10,1,10,9,1,1,15,1,12,11,14,1,18,1,16,1,14,1,
%T 42,1,1,15,20,13,19,1,22,17,20,1,54,1,18,18,26,1,34,1,33,21,20,1,42,
%U 17,22,23,32,1,78,1,34,20,1,19,78,1,24,27,74,1,27,1,40
%N Sum of non-coreful divisors of n.
%C Non-coreful divisor d of a number k is a divisor such that rad(d) != rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
%H Amiram Eldar, <a href="/A308135/b308135.txt">Table of n, a(n) for n = 1..10000</a>
%H G. E. Hardy and M. V. Subbarao, <a href="http://oeis.org/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
%F a(n) = A000203(n) - A057723(n).
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065487 = 0.413642... . - _Amiram Eldar_, Dec 08 2023
%e a(15) = 9. Prime factors of 15 are 3, 5 and its divisors are 1, 3, 5, 15. The non-coreful divisors are 1, 3, 5 and their sum is 9.
%p with(numtheory): P:=proc(k) local a,n; a:=mul(n,n=factorset(k));
%p sigma(k)-a*sigma(k/a); end: seq(P(i),i=1..74);
%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; a[1] = 0; a[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]); Array[a, 100]
%Y Cf. A000203, A005101, A007947, A057723, A307888, A307958, A308029, A308127.
%Y Cf. A013661, A065487.
%K nonn
%O 1,6
%A _Amiram Eldar_ and _Paolo P. Lava_, May 14 2019
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