A129294 -id:A129294 - cp's OEIS frontend

A068601 a(n) = n^3 - 1.

0, 7, 26, 63, 124, 215, 342, 511, 728, 999, 1330, 1727, 2196, 2743, 3374, 4095, 4912, 5831, 6858, 7999, 9260, 10647, 12166, 13823, 15624, 17575, 19682, 21951, 24388, 26999, 29790, 32767, 35936, 39303, 42874, 46655, 50652, 54871, 59318, 63999, 68920

Offset: 1

Naohiro Nomoto, Mar 28 2002

a(n) is the least positive integer k such that k can only contain 'n-1' in exactly 2 different bases B, where 1 < B <= k.
Apart from the first term, the same as A135300. - R. J. Mathar, Apr 29 2008
A058895(n)^3 + a(n)^3 + A033562(n)^3 = A185065(n)^3. - Vincenzo Librandi, Mar 13 2012
Numbers k such that for every nonnegative integer m, k^(3*m+1) + k^(3*m) is a cube. - Arkadiusz Wesolowski, Aug 10 2013

			For n=6; 215 written in bases 6 and 42 is 555, 55 and (555, 55) are exactly 2 different bases.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000217, A003215, A005448, A016921, A033562, A058895, A129294, A135300, A185065, A339604.

List([1..45],n->n^3-1); # Muniru A Asiru, Oct 23 2018

[n^3-1: n in [1..40]]; // Vincenzo Librandi, Mar 11 2012

A068601:=n->n^3-1: seq(A068601(n), n=1..50); # Wesley Ivan Hurt, Jul 23 2014

f[n_]:=n^3-1;f[Range[60]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)
LinearRecurrence[{4,-6,4,-1},{0,7,26,63},50] (* Vincenzo Librandi, Mar 11 2012 *)
Range[50]^3 - 1 (* Wesley Ivan Hurt, Jul 23 2014 *)

PARI
```
a(n)=n^3-1
```

for n in range(1,50): print(n**3-1, end=', ') # Stefano Spezia, Nov 21 2018

Partial sums of A003215, hex (or centered hexagonal) numbers: 3*n(n+1)+1. - Jonathan Vos Post, Mar 16 2006
G.f.: x^2*(7-2*x+x^2)/(1-x)^4. - Colin Barker, Feb 12 2012
4*a(m^2-2*m+2) = (m^2-m+1)^3 + (m^2-m-1)^3 + (m^2-3*m+3)^3 + (m^2-3*m+1)^3. - Bruno Berselli, Jun 23 2014
a(n) = Sum_{i=1..n-1} (i+1)^3 - i^3. - Wesley Ivan Hurt, Jul 23 2014
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(3*n) - 1) = A339604. - Amiram Eldar, Nov 06 2020
Product_{n>=2} (1 + 1/a(n)) = 3*Pi*sech(sqrt(3)*Pi/2). - Amiram Eldar, Jan 20 2021
E.g.f.: 1 + exp(x)*(x^3 + 3*x^2 + x - 1). - Stefano Spezia, Jul 06 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

A129292 Number of divisors of n^4 - 1 that are not greater than n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 4, 5, 3, 8, 3, 10, 4, 6, 4, 12, 4, 13, 4, 11, 6, 14, 3, 10, 6, 12, 6, 17, 3, 16, 7, 10, 9, 13, 4, 18, 7, 11, 4, 22, 3, 26, 8, 9, 7, 23, 5, 18, 7, 13, 6, 25, 4, 24, 8, 21, 6, 18, 3, 18, 10, 12, 14, 16, 4, 26, 8, 17, 7, 31, 5, 30, 6, 11, 13, 26, 7, 25, 6, 16, 10, 35, 4, 18, 11

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

a(n) = #{d: d<=n and A123865(n) mod d = 0};

a(n)>1 for n>2, see A129293 for m such that a(m)=2: a(A129293(n))=2.

			a(100) = #{1,3,9,11,33,73,99} = 7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A129294, A129296.

f:= n -> nops(select(`<=`,numtheory:-divisors(n^4-1),n)):
1,seq(f(n), n=2..100); # Robert Israel, Sep 21 2014

Table[Count[Divisors[n^4-1],?(#<=n&)],{n,90}] (* _Harvey P. Dale, Aug 13 2014 *)

a(n) = if (n==1, 1, sumdiv(n^4-1, d, d <= n)); \\ Michel Marcus, Sep 21 2014

A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 20, 24, 38, 54, 62, 80, 90, 110, 138, 150, 164, 168, 192, 194, 272, 278, 314, 332, 348, 398, 402, 434, 500, 572, 642, 644, 720, 728, 762, 798, 812, 860, 864, 878, 920, 992, 1020, 1022, 1070, 1092, 1098, 1118, 1130, 1182, 1202, 1230, 1260, 1308

Offset: 1

Reinhard Zumkeller, Apr 09 2007

nonn

Numbers m such that A129294(m) = #{1,m-1} = 2.

Essentially the same as A096175. Note that m^3 - 1 = (m - 1)*(m^2 + m + 1), so m - 1 must be prime. For m > 4, the smallest divisor > 1 of m^2 + m + 1 is no larger than sqrt(m^2 + m + 1) < m + 1 unless m^2 + m + 1 is also prime. Also note that gcd(m, m^2 + m 1 ) = gcd(m - 1, m^2 + m + 1) = 1, so m^2 + m + 1 must also be prime, making m^3 - 1 a semiprime. - Jianing Song, Aug 01 2018

			{1,11,157,1727} is the set of divisors of 12^3 - 1, therefore 12 is a term, since A129294(12) = #{1,11} = 2.

Cf. A096175, A129293, A129294.

a(n) = A096175(n-2) for n > 2. - Jianing Song, Aug 01 2018

cp's OEIS Frontend

A068601 a(n) = n^3 - 1.

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

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A129292 Number of divisors of n^4 - 1 that are not greater than n.

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Author

Keywords

Comments

Examples

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Crossrefs

Programs

Maple

Mathematica

PARI

A129295 Numbers m such that m^3 - 1 has no divisors d with 1 < d < m - 1.

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Comments

Examples

Crossrefs

Formula