login
Refactorable numbers that are twice a square.
1

%I #15 Oct 11 2024 13:56:38

%S 2,8,18,72,128,288,450,882,1152,1250,1800,2178,3042,3528,4050,5000,

%T 5202,6498,8712,9522,11250,12168,13122,15138,16200,17298,18432,20808,

%U 24642,25992,28800,30258,32768,33282,38088,39762,45000,50562,52488,56448,60552,62658,64800,66978,69192,71442,80000

%N Refactorable numbers that are twice a square.

%C Intersection of A001105 and A033950.

%H Robert Israel, <a href="/A376114/b376114.txt">Table of n, a(n) for n = 1..10000</a>

%F Conjecture: a(n) = A181795(n)/2.

%e 8 is a term because it is twice 4, which is square, and 8 is refactorable.

%p filter:= proc(n) n mod numtheory:-tau(n) = 0 end proc:

%p select(filter, [seq(2*i^2,i=1..1000)]); # _Robert Israel_, Oct 10 2024

%t Select[Range[2,10^5], IntegerQ@Sqrt[#/2]&&Divisible[#, DivisorSigma[0,#]]&]

%o (PARI) ok(n)=n%numdiv(n)==0&&issquare(n/2)

%o (Python)

%o from itertools import count, islice

%o from math import prod

%o from sympy import factorint

%o def A376114_gen(): # generator of terms

%o for n in count(1):

%o k = prod((e<<1|1)+(p==2) for p, e in factorint(n).items())

%o if not (m:=n**2<<1)%k: yield m

%o A376114_list = list(islice(A376114_gen(),47)) # _Chai Wah Wu_, Oct 04 2024

%Y Cf. A001105, A033950, A181795.

%K nonn

%O 1,1

%A _Waldemar Puszkarz_, Sep 10 2024