A129292 -id:A129292 - cp's OEIS frontend

A123865 a(n) = n^4 - 1.

0, 15, 80, 255, 624, 1295, 2400, 4095, 6560, 9999, 14640, 20735, 28560, 38415, 50624, 65535, 83520, 104975, 130320, 159999, 194480, 234255, 279840, 331775, 390624, 456975, 531440, 614655, 707280, 809999, 923520, 1048575, 1185920, 1336335

Offset: 1

Reinhard Zumkeller, Oct 16 2006

a(n) mod 5 = 0 iff n mod 5 > 0: a(A008587(n)) = 4; a(A047201(n)) = 0; a(n) mod 5 = 4*(1-A079998(n)).

A129292(n) = number of divisors of a(n) that are not greater than n. - Reinhard Zumkeller, Apr 09 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000583, A005563, A024002, A123866, A123867, A123868, A219086.

List([1..40], n-> n^4 -1); # G. C. Greubel, Aug 08 2019

[n^4 - 1: n in [1..40]]; // Vincenzo Librandi, May 01 2011

seq(n^4 -1, n=1..40); # G. C. Greubel, Aug 08 2019

Table[n^4-1,{n,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 15 2011 *)

vector(40, n, n^4 -1) \\ G. C. Greubel, Aug 08 2019

[n^4 -1 for n in (1..40)] # G. C. Greubel, Aug 08 2019

G.f.: x^2*(15 + 5*x + 5*x^2 - x^3)/(1-x)^5. - Colin Barker, Jan 10 2012
-4*a(n+1) = -4*n*(n+2)*(n^2+2*n+2) = (n+n*i)*(n+2+n*i)*(n+(n+2)*i)*(n+2+(n+2)*i), where i is the imaginary unit. - Jon Perry, Feb 05 2014
From Vaclav Kotesovec, Feb 14 2015: (Start)
Sum_{n>=2} 1/a(n) = 7/8 - Pi*coth(Pi)/4 = A256919.
Sum_{n>=2} (-1)^n / a(n) = 1/8 - Pi/(4*sinh(Pi)). (End)
a(n) = A005563(A005563(n)). - Bruno Berselli, May 28 2015
E.g.f.: 1 + (-1 + x + 7*x^2 + 6*x^3 + x^4)*exp(x). - G. C. Greubel, Aug 08 2019
Product_{n>=2} (1 + 1/a(n)) = 4*Pi*csch(Pi). - Amiram Eldar, Jan 20 2021

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

Original entry on oeis.org

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

Reinhard Zumkeller, Apr 09 2007

nonn

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

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

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.

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

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

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

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 *)

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

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

A129293 Numbers m such that m^4-1 has no divisors d with 1 < d < m-1.

Original entry on oeis.org

3, 4, 6, 150, 180, 240, 270, 420, 570, 1290, 1320, 2310, 2550, 2730, 3360, 3390, 4260, 4650, 5850, 5880, 6360, 6780, 9000, 9240, 9630, 10530, 10890, 11970, 13680, 13830, 14010, 14550, 16230, 16650, 18060, 18120, 18540, 19140, 19380, 21600, 21840, 23370

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Essentially the same as A070155, since m^4-1=(m-1)(m+1)(1+m^2). - R. J. Mathar, Jun 14 2008

			{1,5,7,35,37,185,259,1295} is the set of divisors of 6^4-1, therefore 6 is a term, A129292(6) = #{1,3} = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A070155, A129292, A129295, A129297.

is(k) = k == 3 || (isprime(k-1) && isprime(k+1) && isprime(k^2+1)); \\ Amiram Eldar, Apr 15 2024

A129292(a(n)) = #{1, a(n)-1} = 2.

A129294 Number of divisors of n^3 - 1 that are not greater than n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 7, 2, 4, 7, 5, 3, 6, 2, 6, 6, 6, 2, 13, 4, 4, 4, 8, 4, 10, 3, 6, 5, 6, 5, 14, 2, 6, 5, 8, 3, 10, 3, 8, 10, 5, 3, 16, 3, 9, 5, 9, 2, 11, 5, 8, 7, 4, 3, 20, 2, 5, 9, 11, 4, 18, 4, 6, 5, 8, 3, 14, 5, 4, 8, 6, 4, 17, 2, 21, 5, 6, 3, 16, 6, 10, 8, 8, 2, 14, 5, 9, 7, 6, 5, 16

Offset: 2

Reinhard Zumkeller, Apr 09 2007

nonn

a(n) = #{d: d<=n and A068601(n) mod d = 0};

a(n)>1 for n>2, see A129295 for m such that a(m)=2: a(A129295(n))=2.

			a(100) = #{1,3,7,9,11,13,21,27,33,37,39,63,77,91,99} = 15.

R. Zumkeller, Table of n, a(n) for n = 2..1000

Cf. A129292, A129296.

Table[Count[Divisors[n^3-1],?(#<n+1&)],{n,2,100}] (* _Harvey P. Dale, Sep 27 2018 *)

a(n) = sumdiv(n^3-1, d, d <= n); \\ Michel Marcus, Aug 01 2018

a(1)=1 removed by Michel Marcus, Aug 01 2018

cp's OEIS Frontend

A123865 a(n) = n^4 - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A129293 Numbers m such that m^4-1 has no divisors d with 1 < d < m-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A129294 Number of divisors of n^3 - 1 that are not greater than n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions