

A307888


Noncoreful perfect numbers.


4




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 noncoreful divisors of k are considered.


LINKS

G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer. 37 (1983), 277307. (Annotated scanned copy)


FORMULA



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


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



