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

%I A381605 #37 Apr 03 2025 14:03:15
%S A381605 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,
%T A381605 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,
%U A381605 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
%N A381605 Number of distinct prime divisors of n^3+1.
%H A381605 Robert Israel, <a href="/A381605/b381605.txt">Table of n, a(n) for n = 1..10000</a>
%F A381605 a(n) = A001221(A001093(n)).
%F A381605 a(n) > 1 for n > 2. - _Charles R Greathouse IV_, Mar 03 2025
%p A381605 f:= proc(n) nops(numtheory:-factorset(n^3+1)) end proc:
%p A381605 map(f, [$1..50]); # _Robert Israel_, Apr 03 2025
%t A381605 Table[PrimeNu[n^3 + 1], {n, 1, 30}]
%o A381605 (MATLAB)
%o A381605 function result = a(n)
%o A381605 result = numel(unique(factor(sym(n^3+1))));
%o A381605 end
%o A381605 (PARI) a(n) = omega(n^3+1); \\ _Michel Marcus_, Mar 01 2025
%o A381605 (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
%Y A381605 Cf. A001093, A001221, A128428, A366580.
%K A381605 nonn
%O A381605 1,3
%A A381605 _Joost de Winter_, Mar 01 2025