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A376114 Refactorable numbers that are twice a square.

%I A376114 #15 Oct 11 2024 13:56:38
%S A376114 2,8,18,72,128,288,450,882,1152,1250,1800,2178,3042,3528,4050,5000,
%T A376114 5202,6498,8712,9522,11250,12168,13122,15138,16200,17298,18432,20808,
%U A376114 24642,25992,28800,30258,32768,33282,38088,39762,45000,50562,52488,56448,60552,62658,64800,66978,69192,71442,80000
%N A376114 Refactorable numbers that are twice a square.
%C A376114 Intersection of A001105 and A033950.
%H A376114 Robert Israel, <a href="/A376114/b376114.txt">Table of n, a(n) for n = 1..10000</a>
%F A376114 Conjecture: a(n) = A181795(n)/2.
%e A376114 8 is a term because it is twice 4, which is square, and 8 is refactorable.
%p A376114 filter:= proc(n) n mod numtheory:-tau(n) = 0 end proc:
%p A376114 select(filter, [seq(2*i^2,i=1..1000)]); # _Robert Israel_, Oct 10 2024
%t A376114 Select[Range[2,10^5], IntegerQ@Sqrt[#/2]&&Divisible[#, DivisorSigma[0,#]]&]
%o A376114 (PARI) ok(n)=n%numdiv(n)==0&&issquare(n/2)
%o A376114 (Python)
%o A376114 from itertools import count, islice
%o A376114 from math import prod
%o A376114 from sympy import factorint
%o A376114 def A376114_gen(): # generator of terms
%o A376114     for n in count(1):
%o A376114         k = prod((e<<1|1)+(p==2) for p, e in factorint(n).items())
%o A376114         if not (m:=n**2<<1)%k: yield m
%o A376114 A376114_list = list(islice(A376114_gen(),47)) # _Chai Wah Wu_, Oct 04 2024
%Y A376114 Cf. A001105, A033950, A181795.
%K A376114 nonn
%O A376114 1,1
%A A376114 _Waldemar Puszkarz_, Sep 10 2024