OFFSET
1,1
COMMENTS
This sequence is infinite since 6*p^2 is included for all primes p. Terms that are not of the form 6*p^2: 112, 1984, 19900, 32512, 134201344, ...
Includes 4*k if k is an even perfect number: see A000396. - Robert Israel, Jan 06 2019
From Amiram Eldar, Oct 01 2022: (Start)
24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
Equivalently, numbers k such that A325314(k) = -k. (End)
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)
EXAMPLE
24 is a term since A162296(24) = 48 = 2*24.
MAPLE
filter:= proc(n) convert(remove(numtheory:-issqrfree, numtheory:-divisors(n)), `+`)=2*n end proc:
select(filter, [$1..200000]); # Robert Israel, Jan 06 2019
MATHEMATICA
s[1]=0; s[n_] := DivisorSigma[1, n] - Times@@(1+FactorInteger[n][[;; , 1]]); Select[Range[10000], s[#] == 2# &]
PROG
(PARI) s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
isok(n) = s(n) == 2*n; \\ Michel Marcus, Dec 20 2018
(Python)
from sympy import divisors, factorint
A322609_list = [k for k in range(1, 10**3) if sum(d for d in divisors(k, generator=True) if max(factorint(d).values(), default=1) >= 2) == 2*k] # Chai Wah Wu, Sep 19 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Dec 20 2018
STATUS
approved