A381138 a(n) is the number of divisors d of n such that tau(n^(1 + d) + d) = 2^omega(n^(1 + d) + d), where tau = A000005 and omega = A001221.

1, 2, 1, 2, 2, 4, 0, 2, 1, 4, 1, 4, 2, 3, 2, 2, 1, 3, 1, 4, 2, 2, 1, 3, 2, 4, 1, 4, 2, 8, 1, 1, 2, 3, 2, 4, 2, 2, 2, 4, 1, 8, 0, 4, 2, 4, 1, 3, 1, 4, 2, 3, 1, 4, 2, 4, 1, 4, 1, 7, 2, 4, 2, 2, 4, 8, 1, 1, 1, 5, 1, 3, 2, 4, 2, 4, 2, 8, 1, 3, 1, 2, 1, 7, 3, 4, 2

Offset: 1

Author

Juri-Stepan Gerasimov, Feb 15 2025

Keywords

• nonn

Comments

a(n) is the number of divisors d of n such that n^(1 + d) + d is squarefree.

Crossrefs

Cf. A000005, A001221, A005117, A381136.

Programs

Magma

[#[d: d in Divisors(n) | #Divisors(n^(1+d)+d) eq 2^#PrimeDivisors(n^(1+d)+d)]: n in [1..40]];

Mathematica

Table[DivisorSum[n, 1 &, SquareFreeQ[n^(1 + #) + #] &], {n, 50}] (* Michael De Vlieger, Mar 09 2025 *)

PARI

a(n) = sumdiv(n, d, my(f=factor(n^(1+d)+d)); numdiv(f) == 2^omega(f)) \\ Michel Marcus, Feb 19 2025

Extensions

More terms from Jinyuan Wang, Mar 09 2025