A322609 Numbers k such that s(k) = 2*k, where s(k) is the sum of divisors of k that have a square factor (A162296).

24, 54, 112, 150, 294, 726, 1014, 1734, 1984, 2166, 3174, 5046, 5766, 8214, 10086, 11094, 13254, 16854, 19900, 20886, 22326, 26934, 30246, 31974, 32512, 37446, 41334, 47526, 56454, 61206, 63654, 68694, 71286, 76614, 96774, 102966, 112614, 115926, 133206

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

Cf. A162296, A325314.

Subsequence of A005101 and A013929.

Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).

Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.

Sequence in context: A301575 A294156 A108215 * A234238 A228876 A005782

Adjacent sequences: A322606 A322607 A322608 * A322610 A322611 A322612

Keyword

nonn

Author

Amiram Eldar, Dec 20 2018

Status

approved