A396683 a(n) is the number of divisors of A006863(n).

1, 8, 20, 24, 24, 16, 120, 8, 56, 64, 60, 16, 144, 8, 40, 96, 64, 8, 640, 8, 144, 72, 40, 16, 336, 16, 40, 80, 48, 16, 1440, 8, 72, 96, 20, 32, 1536, 8, 20, 48, 336, 16, 720, 8, 96, 256, 40, 8, 768, 8, 160, 48, 48, 16, 1600, 48, 224, 24, 40, 8, 3456, 8, 20, 384, 80, 32, 480, 8

Offset

0,2

Comments

For n > 1, number of k such that psi(k) divides 2*n, where psi = A002322 is the reduced totient function. In other words, number of k such that x^(2*n) == 1 (mod k) for every x coprime to k.

Links

Jianing Song, Table of n, a(n) for n = 0..10000

Formula

a(n) = A000005(A006863(n)).

Prog

(PARI) a(n) = numdiv(A079612(2*n)) \\ See A079612 for its program
(Python)
from math import prod
from sympy import factorint, divisors, isprime
def A396683(n): return prod(e+2 for p, e in factorint(n).items() if p>2 and not 2*n%(p-1))*((~n & n-1).bit_length()+4)<<sum(1 for d in divisors(2*n, generator=True) if d>1 and n%(d+1) and isprime(d+1)) if n else 1 # Chai Wah Wu, Jun 02 2026

Crossrefs

Cf. A006863, A000005, A002322, A395569 (moebius transform).

Sequence in context: A205226 A205318 A100212 * A349168 A083094 A164916

Adjacent sequences: A396679 A396680 A396682 * A396684 A396685 A396686

Keyword

nonn,easy

Author

Jianing Song, Jun 02 2026

Status

approved