A307958 Coreful perfect numbers: numbers k such that csigma(k) = 2*k, where csigma(k) is the sum of the coreful divisors of k (A057723).

36, 180, 252, 392, 396, 468, 612, 684, 828, 1044, 1116, 1176, 1260, 1332, 1476, 1548, 1692, 1908, 1960, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4312, 4572, 4716, 4788

Offset: 1

Author

Amiram Eldar, May 08 2019

Keywords

• nonn

Comments

Hardy and Subbarao defined a coreful divisor d of a number k as a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947). The number of these divisors is A005361(k) and their sum is csigma(k) = A057723(k). Since csigma(k) is multiplicative and csigma(p) = p for prime p, then if k is coreful perfect number, then also m*k is, for any squarefree number m coprime to k, gcd(m, k) = 1. Thus there are infinitely many coreful perfect numbers, and all of them can be generated from the sequence of primitive coreful perfect numbers (A307959), which is the subsequence of powerful terms of this sequence. This sequence and A307959 are analogous to e-perfect numbers (A054979) and primitive e-perfect numbers (A054980).

Examples

			36 is in the sequence since its coreful divisors are 6, 12, 18, 36, whose sum is 72 = 2 * 36.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)

Crossrefs

Cf. A005361, A007947, A054979, A054980, A057723, A307959, A307960, A307961.

Programs

Mathematica

f[p_,e_] := (p^(e+1)-1)/(p-1)-1; a[1]=1; a[n_] := Times @@ (f @@@ FactorInteger[n]); s={}; Do[If[a[n] == 2n, AppendTo[s,n]], {n, 1, 10^6}]; s

PARI

rad(n) = factorback(factorint(n)[, 1]); \\ A007947
s(n) = rad(n)*sigma(n/rad(n)); \\ A057723
isok(n) = s(n) == 2*n; \\ Michel Marcus, May 14 2019