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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).

%I A322609 #34 Oct 01 2022 10:24:48
%S A322609 24,54,112,150,294,726,1014,1734,1984,2166,3174,5046,5766,8214,10086,
%T A322609 11094,13254,16854,19900,20886,22326,26934,30246,31974,32512,37446,
%U A322609 41334,47526,56454,61206,63654,68694,71286,76614,96774,102966,112614,115926,133206
%N 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).
%C A322609 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, ...
%C A322609 Includes 4*k if k is an even perfect number: see A000396. - _Robert Israel_, Jan 06 2019
%C A322609 From _Amiram Eldar_, Oct 01 2022: (Start)
%C A322609 24 = 6*prime(1)^2 = 4*A000396(1) is the only term that is common to the two forms that are mentioned above.
%C A322609 19900 is the only term below 10^11 which is not of any of these two forms. Are there any other such terms?
%C A322609 All the known nonunitary perfect numbers (A064591) are also of the form 4*k, where k is an even perfect number.
%C A322609 Equivalently, numbers k such that A325314(k) = -k. (End)
%H A322609 Amiram Eldar, <a href="/A322609/b322609.txt">Table of n, a(n) for n = 1..12091</a> (terms below 10^11; terms 1..300 from Robert Israel)
%e A322609 24 is a term since A162296(24) = 48 = 2*24.
%p A322609 filter:= proc(n) convert(remove(numtheory:-issqrfree,numtheory:-divisors(n)),`+`)=2*n end proc:
%p A322609 select(filter, [$1..200000]); # _Robert Israel_, Jan 06 2019
%t A322609 s[1]=0; s[n_] := DivisorSigma[1,n] - Times@@(1+FactorInteger[n][[;;,1]]); Select[Range[10000], s[#] == 2# &]
%o A322609 (PARI) s(n) = sumdiv(n, d, d*(1-moebius(d)^2)); \\ A162296
%o A322609 isok(n) = s(n) == 2*n; \\ _Michel Marcus_, Dec 20 2018
%o A322609 (Python)
%o A322609 from sympy import divisors, factorint
%o A322609 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
%Y A322609 Cf. A162296, A325314.
%Y A322609 Subsequence of A005101 and A013929.
%Y A322609 Numbers k such that A162296(k) = m*k: A005117 (m=0), A001248 (m=1), this sequence (m=2), A357493 (m=3), A357494 (m=4).
%Y A322609 Similar sequences: A000396, A002827, A007357, A054979, A064591, A322486, A323757, A324707, A327633.
%K A322609 nonn
%O A322609 1,1
%A A322609 _Amiram Eldar_, Dec 20 2018