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

6, 1638, 55860, 168836850, 12854283750

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).

Sequence is a subset of A083207.

Tested up to 10^12. - Giovanni Resta, May 10 2019

Links

Table of n, a(n) for n=1..5.

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000203, A007947, A057723, A083207, A307888, A307958, A307962, A307963, A307986.

Sequence in context: A378241 A281255 A216934 * A160226 A209609 A350595

Adjacent sequences: A308026 A308027 A308028 * A308030 A308031 A308032

Keyword

nonn,more

Author

Paolo P. Lava, May 10 2019

Extensions

a(4)-a(5) from Giovanni Resta, May 10 2019

Status

approved