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A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.

%I A308029 #19 May 12 2019 03:08:53
%S A308029 6,1638,55860,168836850,12854283750
%N A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.
%C A308029 A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
%C A308029 Sequence is a subset of A083207.
%C A308029 Tested up to 10^12. - _Giovanni Resta_, May 10 2019
%H A308029 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 A308029 Solutions of A000203(k) = 2*A057723(k).
%e A308029 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 A308029 with(numtheory): P:=proc(q) local a, k, n; for n from 1 to q do
%p A308029 a:=mul(k, k=factorset(n)); if sigma(n)=2*a*sigma(n/a)
%p A308029 then print(n); fi; od; end: P(10^7);
%t A308029 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 A308029 (PARI) rad(n) = factorback(factorint(n)[, 1]); \\ A007947
%o A308029 s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
%o A308029 isok(n) = 2*s(n) == sigma(n); \\ _Michel Marcus_, May 11 2019
%Y A308029 Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986.
%K A308029 nonn,more
%O A308029 1,1
%A A308029 _Paolo P. Lava_, May 10 2019
%E A308029 a(4)-a(5) from _Giovanni Resta_, May 10 2019