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

1,3

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

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.

Formula

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

Example

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

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 *)

Prog

(PARI) a(n) = if (n==1, 1, sumdiv(n^2-1, d, d<=n)); \\ Michel Marcus, Jan 02 2014
(Haskell)
a129296 n = length [d | d <- [1..n], (n ^ 2 - 1) `mod` d == 0]
-- Reinhard Zumkeller, Jan 09 2014

Crossrefs

Cf. A000005, A005563, A129292, A129294.

Sequence in context: A328720 A005128 A187782 * A300837 A321443 A333836

Adjacent sequences: A129293 A129294 A129295 * A129297 A129298 A129299

Keyword

nonn

Author

Reinhard Zumkeller, Apr 09 2007

Status

approved