A082863 Number of distinct prime factors of n^2-1.

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

Offset: 2

Author

Jon Perry, May 24 2003

Keywords

• nonn

Comments

This is a very slowly growing sequence - by n=100000 the maximum value is 8.

If n is in A014574 then a(n) = 2. - Robert Israel, Aug 05 2014

Examples

			a(11)=3 because (11-1)*(11+1)=10*12=2^3*3*5, which has 3 distinct prime factors, namely 2,3 and 5.
		

Links

• T. D. Noe, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001221, A014574, A219017 (greedy inverse).

Programs

Maple

A082863 := proc(n)
        A001221(n^2-1) ;
end proc: # R. J. Mathar, Aug 05 2014
# alternative:
A082863:= n -> nops(numtheory:-factorset(n^2-1)):
seq(A082863(n), n=1..100); # Robert Israel, Aug 05 2014

Mathematica

Table[PrimeNu[n^2-1],{n,2,100}] (* Harvey P. Dale, Jul 05 2011 *)

PARI

for (n=2,100,print1(omega((n-1)*(n+1))","))

Formula

a(n) = A001221((n-1)*(n+1)).

a(n) = A001221(n-1) + A001221(n+1) + ((-1)^n - 1)/2. - Robert Israel, Aug 05 2014