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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). 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 coprime m, 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).

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)

EXAMPLE

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

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

PROG

(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

CROSSREFS

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

Sequence in context: A211751 A211762 A219858 * A127657 A318100 A335218

Adjacent sequences:  A307955 A307956 A307957 * A307959 A307960 A307961

KEYWORD

nonn

AUTHOR

Amiram Eldar, May 08 2019

STATUS

approved

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Last modified June 23 23:00 EDT 2021. Contains 345402 sequences. (Running on oeis4.)