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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 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

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