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%I #19 May 12 2019 03:08:53
%S 6,1638,55860,168836850,12854283750
%N Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.
%C A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
%C Sequence is a subset of A083207.
%C Tested up to 10^12. - _Giovanni Resta_, May 10 2019
%H G. E. Hardy and M. V. Subbarao, <a href="/A005934/a005934.pdf">Highly powerful numbers</a>, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)
%F Solutions of A000203(k) = 2*A057723(k).
%e Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.
%p with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
%p a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
%p then print(n); fi; od; end: P(10^7);
%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* _Amiram Eldar_, May 11 2019 *)
%o (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
%o isok(n) = 2*s(n) == sigma(n); \\ _Michel Marcus_, May 11 2019
%Y Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986.
%K nonn,more
%O 1,1
%A _Paolo P. Lava_, May 10 2019
%E a(4)-a(5) from _Giovanni Resta_, May 10 2019
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