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A307888 Non-coreful perfect numbers. 4
6, 234, 588, 600, 6552, 89376, 209195610624 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A286392 A221926 A324232 * A194482 A309330 A362733
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, May 09 2019
EXTENSIONS
a(7) from Giovanni Resta, May 09 2019
STATUS
approved

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Last modified July 25 02:49 EDT 2024. Contains 374585 sequences. (Running on oeis4.)