A396683 - OEIS

%I #18 Jun 03 2026 00:51:14

%S 1,8,20,24,24,16,120,8,56,64,60,16,144,8,40,96,64,8,640,8,144,72,40,

%T 16,336,16,40,80,48,16,1440,8,72,96,20,32,1536,8,20,48,336,16,720,8,

%U 96,256,40,8,768,8,160,48,48,16,1600,48,224,24,40,8,3456,8,20,384,80,32,480,8

%N a(n) is the number of divisors of A006863(n).

%C 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.

%H Jianing Song, <a href="/A396683/b396683.txt">Table of n, a(n) for n = 0..10000</a>

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

%o (PARI) a(n) = numdiv(A079612(2*n)) \\ See A079612 for its program

%o (Python)

%o from math import prod

%o from sympy import factorint, divisors, isprime

%o 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

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

%K nonn,easy

%O 0,2

%A _Jianing Song_, Jun 02 2026