cp's OEIS Frontend

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

%I A129296 #45 Dec 03 2024 12:23:01
%S A129296 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,
%T A129296 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,
%U A129296 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
%N A129296 Number of divisors of n^2 - 1 that are not greater than n.
%C A129296 a(n) = #{d: d<=n and A005563(n+1) mod d = 0};
%C A129296 a(n)>1 for n>2, see A129297 for m such that a(m)=2: a(A129297(n)) = 2.
%C A129296 If a(6n) = 2 for n>=1, then 6n-1 and 6n+1 are twin primes see A129297. - _Fred Daniel Kline_, Jan 02 2014
%C A129296 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
%H A129296 Reinhard Zumkeller, <a href="/A129296/b129296.txt">Table of n, a(n) for n = 1..10000</a>
%H A129296 Adrian W. Dudek, <a href="https://doi.org/10.1017/S0004972715001136">On the number of divisors of n^2-1</a>, Bulletin of the Australian Mathematical Society, Vol. 93, No. 2 (2016), pp. 194-198; <a href="https://arxiv.org/abs/1507.08893">arXiv preprint</a>, arXiv:1507.08893 [math.NT], 2015.
%F A129296 a(n) = A000005(n^2-1)/2 for n >= 2. - _Robert Israel_, Aug 03 2015
%e A129296 a(100) = #{1,3,9,11,33,99} = 6.
%p A129296 1,seq(numtheory:-tau(n^2-1)/2,n=2..100); # _Robert Israel_, Aug 03 2015
%t A129296 nd[n_]:=Count[Divisors[n^2-1],_?(#<=n&)]; Array[nd,100] (* _Harvey P. Dale_, Jan 03 2014 *)
%t A129296 a[n_] := DivisorSigma[0, n^2 -1]/2; a[1] = 1; Array[a, 100] (* _Amiram Eldar_, Jun 18 2022 *)
%o A129296 (PARI) a(n) = if (n==1, 1, sumdiv(n^2-1, d, d<=n)); \\ _Michel Marcus_, Jan 02 2014
%o A129296 (Haskell)
%o A129296 a129296 n = length [d | d <- [1..n], (n ^ 2 - 1) `mod` d == 0]
%o A129296 -- _Reinhard Zumkeller_, Jan 09 2014
%Y A129296 Cf. A000005, A005563, A063647, A129292, A129294.
%K A129296 nonn
%O A129296 1,3
%A A129296 _Reinhard Zumkeller_, Apr 09 2007