A129296 Number of divisors of n^2 - 1 that are not greater than n.

1, 1, 2, 2, 4, 2, 5, 3, 5, 3, 8, 2, 8, 4, 6, 4, 9, 2, 12, 4, 8, 4, 10, 3, 10, 6, 8, 4, 16, 2, 14, 4, 7, 8, 12, 4, 12, 4, 10, 4, 20, 2, 16, 6, 8, 6, 12, 3, 18, 6, 12, 4, 16, 4, 20, 8, 10, 4, 16, 2, 16, 6, 8, 12, 16, 4, 16, 4, 16, 4, 30, 2, 15, 6, 8, 12, 16, 4, 24, 5, 12, 5, 16, 4, 16, 8, 10, 4, 30, 4

Offset: 1

Author

Reinhard Zumkeller, Apr 09 2007

Keywords

• nonn

Comments

a(n) = #{d: d<=n and A005563(n+1) mod d = 0};

a(n)>1 for n>2, see A129297 for m such that a(m)=2: a(A129297(n)) = 2.

If a(6n) = 2 for n>=1, then 6n-1 and 6n+1 are twin primes see A129297. - Fred Daniel Kline, Jan 02 2014

For n>1, a(n) is the number of positive integers k such that k+n divides k*n+1. - Thomas Ordowski, Dec 01 2024

Examples

			a(100) = #{1,3,9,11,33,99} = 6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Adrian W. Dudek, On the number of divisors of n^2-1, Bulletin of the Australian Mathematical Society, Vol. 93, No. 2 (2016), pp. 194-198; arXiv preprint, arXiv:1507.08893 [math.NT], 2015.

Crossrefs

Cf. A000005, A005563, A063647, A129292, A129294.

Programs

Haskell

a129296 n = length [d | d <- [1..n], (n ^ 2 - 1) `mod` d == 0]
-- Reinhard Zumkeller, Jan 09 2014

Maple

1,seq(numtheory:-tau(n^2-1)/2,n=2..100); # Robert Israel, Aug 03 2015

Mathematica

nd[n_]:=Count[Divisors[n^2-1],?(#<=n&)]; Array[nd,100] (* _Harvey P. Dale, Jan 03 2014 *)
a[n_] := DivisorSigma[0, n^2 -1]/2; a[1] = 1; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

PARI

a(n) = if (n==1, 1, sumdiv(n^2-1, d, d<=n)); \\ Michel Marcus, Jan 02 2014

Formula

a(n) = A000005(n^2-1)/2 for n >= 2. - Robert Israel, Aug 03 2015