A129296 - OEIS

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

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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)