A307888 - OEIS

%I #58 May 14 2019 11:02:00

%S 6,234,588,600,6552,89376,209195610624

%N Non-coreful perfect numbers.

%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 Here, only the non-coreful divisors of k are considered.

%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 k = A000203(k) - A057723(k).

%e Divisors of 234 are 1, 2, 3, 6, 9, 13, 18, 26, 39, 78, 117, 234 and its prime factors are 2, 3, 13. Among the divisors, 78 and 234 are divided by all the prime factors and 1 + 2 + 3 + 6 + 9 + 13 + 18 + 26 + 39 + 117 = 234.

%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 n=sigma(n)-a*sigma(n/a) then print(n); fi;

%p od; end: P(10^7);

%t f[p_, e_] := (p^(e + 1) - 1)/(p - 1); fc[p_, e_] := f[p, e] - 1; ncQ[n_] := Times @@ (f @@@ FactorInteger[n]) - Times @@ (fc @@@ FactorInteger[n]) == n; Select[Range[2, 10^5], ncQ] (* _Amiram Eldar_, May 11 2019 *)

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

%o s(n) = rad(n)*sigma(n/rad(n)); \\ A057723

%o isok(n) = sigma(n) - s(n) == n; \\ _Michel Marcus_, May 11 2019

%Y Cf. A000203, A007947, A057723, A307958, A307986, A308029.

%K nonn,more

%O 1,1

%A _Paolo P. Lava_, May 09 2019

%E a(7) from _Giovanni Resta_, May 09 2019