A359580 Numbers that are either (an odd squarefree number squared) or twice such a number.

1, 2, 9, 18, 25, 49, 50, 98, 121, 169, 225, 242, 289, 338, 361, 441, 450, 529, 578, 722, 841, 882, 961, 1058, 1089, 1225, 1369, 1521, 1681, 1682, 1849, 1922, 2178, 2209, 2450, 2601, 2738, 2809, 3025, 3042, 3249, 3362, 3481, 3698, 3721, 4225, 4418, 4489, 4761, 5041, 5202, 5329, 5618, 5929, 6050, 6241

Offset: 1

Author

Antti Karttunen, Jan 07 2023

Keywords

• nonn

Comments

Numbers in whose prime factorization the exponent of 2 can be only 0 or 1, and the exponent of any odd prime can be only 0 or 2.

Crossrefs

Cf. A359549 (characteristic function).

Positions of odd terms in A046692, A327276, A327278, A359548.

Cf. also A056911, A062503.

Programs

Mathematica

Select[Range[6000], (e = IntegerExponent[#, 2]) < 2 && SquareFreeQ[Sqrt[#/2^e]] &] (* Amiram Eldar, Jan 07 2023 *)
{#,2#}&/@(Select[Range[1,101,2],SquareFreeQ]^2)//Flatten//Sort (* Harvey P. Dale, May 27 2025 *)

PARI

isA359580(n) = A359549(n);

Python

from itertools import count, islice
from sympy import factorint
def A359580_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n & n-1).bit_length())<=1 and all(e==2 for e in factorint(n>>m).values()),count(max(startvalue,1)))
A359580_list = list(islice(A359580_gen(),20)) # Chai Wah Wu, Jan 11 2023

Formula

Sum_{n>=1} 1/a(n) = 18/Pi^2. - Amiram Eldar, Jan 07 2023