A307888 Non-coreful perfect numbers.

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

Offset: 1

Author

Paolo P. Lava, May 09 2019

Keywords

• nonn
• more

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).

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

Examples

			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.
		

Links

• G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277-307. (Annotated scanned copy)

Crossrefs

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

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n; for n from 1 to q do
a:=mul(k,k=factorset(n)); if n=sigma(n)-a*sigma(n/a) then print(n); fi;
od; end: P(10^7);

Mathematica

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 *)

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(n) = sigma(n) - s(n) == n; \\ Michel Marcus, May 11 2019

Formula

Solutions of k = A000203(k) - A057723(k).

Extensions

a(7) from Giovanni Resta, May 09 2019