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

Author

Amiram Eldar, Dec 20 2018

Keywords

• nonn

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)

Examples

			24 is a term since A162296(24) = 48 = 2*24.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..12091 (terms below 10^11; terms 1..300 from Robert Israel)

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.

Programs

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# &]

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