cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A307888 Non-coreful perfect numbers.

Original entry on oeis.org

6, 234, 588, 600, 6552, 89376, 209195610624
Offset: 1

Views

Author

Paolo P. Lava, May 09 2019

Keywords

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

A308029 Numbers whose sum of coreful divisors is equal to the sum of non-coreful divisors.

Original entry on oeis.org

6, 1638, 55860, 168836850, 12854283750
Offset: 1

Views

Author

Paolo P. Lava, May 10 2019

Keywords

Comments

A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k (see LINKS).
Sequence is a subset of A083207.
Tested up to 10^12. - Giovanni Resta, May 10 2019

Examples

			Divisors of 1638 are 1, 2, 3, 6, 7, 9, 13, 14, 18, 21, 26, 39, 42, 63, 78, 91, 117, 126, 182, 234, 273, 546, 819, 1638. The coreful ones are 546, 1638 and 1 + 2 + 3 + 6 + 7 + 9 + 13 + 14 + 18 + 21 + 26 + 39 + 42 + 63 + 78 + 91 + 117 + 126 + 182 + 234 + 273 + 819 = 546 + 1638 = 2184.

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, A083207, A307888, A307958, A307962, A307963, A307986.

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 sigma(n)=2*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; csigmaQ[n_] := Times @@ (fc @@@ FactorInteger[n]) == Times @@ (f @@@ FactorInteger[n])/2; Select[Range[2, 10^5], csigmaQ] (* Amiram Eldar, May 11 2019 *)
  • PARI
    rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = my(rn=rad(n)); rn*sigma(n/rn); \\ A057723
isok(n) = 2*s(n) == sigma(n); \\ Michel Marcus, May 11 2019

Formula

Solutions of A000203(k) = 2*A057723(k).

Extensions

a(4)-a(5) from Giovanni Resta, May 10 2019
Showing 1-2 of 2 results.

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