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

6, 1638, 55860, 168836850, 12854283750

Offset: 1

Author

Paolo P. Lava, May 10 2019

Keywords

• nonn
• more

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