A355058 Numbers m such that d(m) mod 6 = 3, where d(m) is the number of divisors of m.

4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364

Offset: 1

Author

Michael De Vlieger, Jul 04 2022

Keywords

• nonn
• easy

Comments

All terms are square; contains squares of primes.

Examples

			Let p be a prime; p^2 has 3 divisors {1, p, p^2}, therefore all squares of primes {4, 9, 25, 49, ...} are in the sequence.
36 is in the sequence because d(36) = 9, and 9 mod 6 = 3.
16 is not in the sequence because it has 5 divisors, and 5 mod 6 = 5.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000290, A001248, A086463, A352475.

Programs

Mathematica

Select[Range[2^12], Mod[DivisorSigma[0, #], 6] == 3 &]

PARI

isok(m) = (numdiv(m) % 6) == 3; \\ Michel Marcus, Jul 05 2022

Python

from itertools import count, islice
from sympy import factorint, prod
def A355058_gen(): # generator of terms
    return map(lambda n:n**2,filter(lambda n:prod((2*e+1)%6 for e in factorint(n).values())%6==3,count(1)))
A355058_list = list(islice(A355058_gen(),30)) # Chai Wah Wu, Jul 06 2022

Formula

{ A000290 \ A352475 }.

Sum_{n>=1} 1/a(n) = Pi^2/18 (A086463). - Amiram Eldar, Jul 06 2022