A376114 Refactorable numbers that are twice a square.

2, 8, 18, 72, 128, 288, 450, 882, 1152, 1250, 1800, 2178, 3042, 3528, 4050, 5000, 5202, 6498, 8712, 9522, 11250, 12168, 13122, 15138, 16200, 17298, 18432, 20808, 24642, 25992, 28800, 30258, 32768, 33282, 38088, 39762, 45000, 50562, 52488, 56448, 60552, 62658, 64800, 66978, 69192, 71442, 80000

Offset: 1

Author

Waldemar Puszkarz, Sep 10 2024

Keywords

• nonn

Comments

Intersection of A001105 and A033950.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001105, A033950, A181795.

Programs

Maple

filter:= proc(n) n mod numtheory:-tau(n) = 0 end proc:
select(filter, [seq(2*i^2,i=1..1000)]); # Robert Israel, Oct 10 2024

Mathematica

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

PARI

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

Python

from itertools import count, islice
from math import prod
from sympy import factorint
def A376114_gen(): # generator of terms
    for n in count(1):
        k = prod((e<<1|1)+(p==2) for p, e in factorint(n).items())
        if not (m:=n**2<<1)%k: yield m
A376114_list = list(islice(A376114_gen(),47)) # Chai Wah Wu, Oct 04 2024

Formula

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