A381605 Number of distinct prime divisors of n^3+1.

1, 1, 2, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 4, 2, 3, 3, 3, 2, 3, 3, 3, 3, 4, 2, 4, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 2, 4, 3, 4, 3, 4, 3, 4, 4, 3, 3, 3, 4, 3, 4, 4, 2, 5, 2, 4, 4, 2, 4, 5, 3, 3, 4, 5, 2, 3, 2, 4, 3, 5, 3, 4, 2, 3, 3, 3, 4, 4, 4, 4, 3, 4, 4, 5, 3, 3, 3, 4, 4, 4, 3, 4, 3, 4, 2

Offset: 1

Author

Joost de Winter, Mar 01 2025

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001093, A001221, A128428, A366580.

Programs

MATLAB

function result = a(n)
result = numel(unique(factor(sym(n^3+1))));
end

Maple

f:= proc(n) nops(numtheory:-factorset(n^3+1)) end proc:
map(f, [$1..50]); # Robert Israel, Apr 03 2025

Mathematica

Table[PrimeNu[n^3 + 1], {n, 1, 30}]

PARI

a(n) = omega(n^3+1); \\ Michel Marcus, Mar 01 2025

PARI

a(n)=if(n%3==2, omega((n+1)/3^valuation(n+1,3)) + omega((n^2-n+1)/3^valuation(n^2-n+1,3)) + 1, omega(n+1) + omega(n^2-n+1)) \\ Charles R Greathouse IV, Mar 03 2025

Formula

a(n) = A001221(A001093(n)).

a(n) > 1 for n > 2. - Charles R Greathouse IV, Mar 03 2025