A307888 Non-coreful perfect numbers.

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

Offset

1,1

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.

Links

Table of n, a(n) for n=1..7.

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A286392 A221926 A324232 * A194482 A309330 A362733

Adjacent sequences: A307885 A307886 A307887 * A307889 A307890 A307891

Keyword

nonn,more

Author

Paolo P. Lava, May 09 2019

Extensions

a(7) from Giovanni Resta, May 09 2019

Status

approved