A024049 -id:A024049 - cp's OEIS frontend

A074479 Largest prime factor of 5^n - 1.

2, 3, 31, 13, 71, 31, 19531, 313, 829, 521, 12207031, 601, 305175781, 19531, 1741, 11489, 466344409, 5167, 3981071, 9161, 519499, 12207031, 332207361361, 390001, 9384251, 305175781, 31051, 234750601, 22125996444329, 7621

Offset: 1

Rick L. Shepherd, Aug 23 2002

nonn

			5^9 - 1 = 1953124 = (2^2)*19*31*829, so a(9) = 829.

Table of n, a(n) for n = 1..502
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A074478 (largest prime factor of 5^n + 1), A074477 (largest prime factor of 3^n - 1), A074249 (largest prime factor of 7^n - 1).
Cf. A006530, A024049.
Cf. similar sequences listed in A274906.

[Maximum(PrimeDivisors(5^n-1)): n in [1..45]]; // Vincenzo Librandi, Jul 13 2016

Table[FactorInteger[5^n - 1] [[-1, 1]], {n, 30}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=1,32, v=factor(5^n-1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A024049(n)). - Vincenzo Librandi, Jul 13 2016

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(448) in b-file from Amiram Eldar, Feb 01 2020
a(449)-a(502) in b-file from Max Alekseyev, Apr 25 2022

A366613 Sum of the divisors of 5^n-1.

Original entry on oeis.org

7, 60, 224, 1736, 6048, 49920, 136724, 1107792, 3718400, 27060480, 85449224, 869499904, 2136230474, 15820920000, 61359427584, 461863805760, 1338408456700, 13177159680000, 33558717136896, 301282248701952, 863701914880000, 6313641012910080, 20863951122979048

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=224 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A000203, A057956, A059887, A074479, A143665, A218357, A295502, A366611, A366612.

Cf. A075708, A366576, A366603, A366622, A366634, A366653, A366662, A102146, A366684, A366710.

a:=n->numtheory[sigma](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[1, 5^Range[30]-1]
```

a(n) = sigma(5^n-1) = A000203(A024049(n)).

A024140 a(n) = 12^n - 1.

Original entry on oeis.org

0, 11, 143, 1727, 20735, 248831, 2985983, 35831807, 429981695, 5159780351, 61917364223, 743008370687, 8916100448255, 106993205379071, 1283918464548863, 15407021574586367, 184884258895036415

Offset: 0

N. J. A. Sloane

In base 12 these are 0, B, BB, BBB, ... . - David Rabahy, Dec 12 2016

Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A001021, A016125, A178248.

Cf. Similar sequences of the type k^n-1: A000004 (k=1), A000225 (k=2), A024023 (k=3), A024036 (k=4), A024049 (k=5), A024062 (k=6), A024075 (k=7), A024088 (k=8), A024101 (k=9), A002283 (k=10), A024127 (k=11), this sequence (k=12).

12^Range[0,20]-1 (* or *) LinearRecurrence[{13,-12},{0,11},20] (* Harvey P. Dale, Feb 01 2019 *)

From Mohammad K. Azarian, Jan 14 2009: (Start)
G.f.: 1/(1-12*x) - 1/(1-x).
E.g.f.: exp(12*x) - exp(x). (End)
a(n) = 12*a(n-1) + 11 for n>0, a(0)=0. - Vincenzo Librandi, Nov 18 2010
a(n) = Sum_{i=1..n} 11^i*binomial(n,n-i) for n>0, a(0)=0. - Bruno Berselli, Nov 11 2015
From Elmo R. Oliveira, Dec 15 2023: (Start)
a(n) = 13*a(n-1) - 12*a(n-2) for n>1.
a(n) = A001021(n)-1 = A178248(n)-2.
a(n) = 11*(A016125(n) - 1)/12. (End)

A366612 Number of divisors of 5^n-1.

Original entry on oeis.org

3, 8, 6, 20, 12, 48, 6, 48, 24, 64, 6, 240, 6, 64, 96, 224, 12, 512, 24, 640, 48, 128, 12, 1152, 192, 64, 384, 320, 24, 6144, 12, 1024, 48, 128, 384, 10240, 24, 512, 48, 6144, 12, 18432, 12, 1280, 3072, 128, 6, 10752, 12, 4096, 192, 960, 24, 81920, 576, 1536

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

			a(3)=6 because 5^3-1 has divisors {1, 2, 4, 31, 62, 124}.

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A000005, A057956, A059887, A074479, A143665, A295502, A218357, A366611, A366613.

Cf. A046801, A366575, A366602, A366621, A366633, A366652, A366661, A070528, A366683, A366709.

a:=n->numtheory[tau](5^n-1):
seq(a(n), n=1..100);

Mathematica
```
DivisorSigma[0, 5^Range[100]-1]
```
PARI
```
a(n) = numdiv(5^n-1);
```

a(n) = sigma0(5^n-1) = A000005(A024049(n)).

A366611 Number of distinct prime divisors of 5^n - 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 5, 2, 6, 2, 5, 6, 6, 3, 7, 4, 8, 5, 6, 3, 8, 7, 5, 8, 7, 4, 11, 3, 8, 5, 6, 8, 11, 4, 8, 5, 11, 3, 12, 3, 9, 11, 6, 2, 11, 3, 11, 7, 8, 4, 14, 8, 9, 6, 7, 3, 17, 4, 7, 10, 11, 7, 12, 6, 11, 8, 14, 3, 16, 4, 8, 15, 11, 6, 11, 4, 15

Offset: 1

Sean A. Irvine, Oct 14 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502

Cf. A024049, A001221, A057956, A059887, A074479, A143665, A218357, A295502, A366612, A366613.

Cf. A046800, A133801, A366604, A366620, A366632, A366651, A366660, A102347, A366681, A366707.

for(n = 1, 100, print1(omega(5^n - 1), ", "))

a(n) = omega(5^n-1) = A001221(A024049(n)).

A125831 a(n) = (5^n - 1)/2.

Original entry on oeis.org

0, 2, 12, 62, 312, 1562, 7812, 39062, 195312, 976562, 4882812, 24414062, 122070312, 610351562, 3051757812, 15258789062, 76293945312, 381469726562, 1907348632812, 9536743164062, 47683715820312, 238418579101562, 1192092895507812, 5960464477539062, 29802322387695312

Offset: 0

Zerinvary Lajos, Feb 03 2007

Number of compositions of odd numbers into n parts < 5. - Adi Dani, Jun 11 2011

Numbers whose base 5 representation is 22222...2 (n times).

			a(2)=12: there are 12 compositions of odd numbers into 2 parts < 5:
1: (0,1),(1,0);
3: (0,3),(3,0),(1,2),(2,1);
5: (1,4),(4,1),(2,3),(3,2);
7: (3,4),(4,3). - _Adi Dani_, Jun 11 2011

S. J. Cyvin, B. N. Cyvin, and J. Brunvoll. Enumeration of tree-like octagonal systems: catapolyoctagons, ACH Models in Chem. 134 (1997), pp. 55-70, eqs. (6) and (7) on p. 58.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Adi Dani, Restricted compositions of natural numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A003463, A024049, A121177 (same with different offset).

List([0..30], n-> (5^n-1)/2); # G. C. Greubel, Aug 03 2019

[(5^n-1)/2: n in [0..30]]; // Vincenzo Librandi, Jun 11 2011

Maple
```
seq((5^n-1)/2, n=0..30);
```

Table[(5^n -1)/2, {n, 0, 30}] (* Harvey P. Dale, Dec 03 2010 *)

a(n)=5^n\2 \\ Charles R Greathouse IV, Jun 11 2011

[(5^n-1)/2 for n in (0..30)] # G. C. Greubel, Aug 03 2019

a(n) = 5*a(n-1) + 2 for n > 0, a(0)=0. - Vincenzo Librandi, Sep 30 2010
From Colin Barker, May 16 2013: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: 2*x/((1-x)*(1-5*x)). (End)
a(n) = 2*A003463(n). - Joerg Arndt, Aug 03 2019
From Elmo R. Oliveira, Dec 10 2023: (Start)
a(n) = A024049(n)/2.
E.g.f.: (1/2)*(exp(5*x) - exp(x)). (End)

Offset corrected by N. J. A. Sloane, Oct 02 2010

Major edit by Joerg Arndt, Jun 11 2011

A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 80, 81, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 97, 98, 99, 101, 103, 105, 109, 110, 111, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 127, 128

Offset: 1

Antti Karttunen, Nov 02 2014

nonn

Integers n such that A249151(n) < n.

Equally: Integers n such that A249431(n) is negative.

			See the examples at A249434.

Antti Karttunen, Table of n, a(n) for n = 1..2676

Complement: A249434.
Subsequences: A000225, A024023, A024049, etc., (after their two initial terms, i.e. A249435 without its initial zero is also a subsequence), A249424, A249436.
Cf. A000142, A001142, A007318, A249151, A249431.

A258807 a(n) = n^5 - 1.

Original entry on oeis.org

0, 31, 242, 1023, 3124, 7775, 16806, 32767, 59048, 99999, 161050, 248831, 371292, 537823, 759374, 1048575, 1419856, 1889567, 2476098, 3199999, 4084100, 5153631, 6436342, 7962623, 9765624, 11881375, 14348906, 17210367, 20511148, 24299999, 28629150, 33554431

Offset: 1

Vincenzo Librandi, Jun 11 2015

Muniru A Asiru, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Subsequence of A181124.
Cf. A024003, A024049, A300656.
Sequences of the type n^k-1: A132411 (k=2), A068601 (k=3), A123865 (k=4), this sequence (k=5), A123866 (k=6), A258808 (k=7), A258809 (k=8), A258810 (k=9), A123867 (k=10), A258812 (k=11), A123868 (k=12).

List([1..35],n->n^5-1); # Muniru A Asiru, Oct 28 2018

Magma
```
[n^5-1: n in [1..50]];
```

I:=[0,31,242,1023, 3124,7775]; [n le 6 select I[n] else 6*Self(n-1)-15*Self(n-2)+20*Self(n-3)-15*Self(n-4)+ 6*Self(n-5)-Self(n-6): n in [1..50]];

seq(n^5-1,n=1..35); # Muniru A Asiru, Oct 28 2018

Table[n^5 - 1, {n, 1, 50}] (* or *) LinearRecurrence[{6, -15, 20, -15, 6, -1}, {0, 31, 242, 1023, 3124, 7775}, 50]

a(n)=n^5-1 \\ Charles R Greathouse IV, Jun 11 2015

for n in range(1, 50): print(n**5 - 1, end=', ') # Stefano Spezia, Oct 28 2018

[n^5-1 for n in (1..50)] # Bruno Berselli, Jun 11 2015

G.f.: x^2*(31 + 56*x + 36*x^2 - 4*x^3 + x^4)/(1 - x)^6.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = -A024003(n). - Bruno Berselli, Jun 11 2015
Sum_{n>=2} 1/a(n) = Sum_{n>=1} (zeta(5*n) - 1) = 0.0379539032... - Amiram Eldar, Nov 06 2020

A024003 a(n) = 1 - n^5.

Original entry on oeis.org

1, 0, -31, -242, -1023, -3124, -7775, -16806, -32767, -59048, -99999, -161050, -248831, -371292, -537823, -759374, -1048575, -1419856, -1889567, -2476098, -3199999, -4084100, -5153631, -6436342, -7962623, -9765624, -11881375, -14348906, -17210367, -20511148

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..555
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A024049.

[1-n^5: n in [0..50]]; // Vincenzo Librandi, Apr 29 2011

1-Range[0,50]^5 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
CoefficientList[Series[(1-6*x-16*x^2-76*x^3-21*x^4-2*x^5)/(1-x)^6, {x, 0, 50}], x] (* G. C. Greubel, May 11 2017 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,0,-31,-242,-1023,-3124},30] (* Harvey P. Dale, May 18 2019 *)

x='x+O('x^50); Vec((1-6*x-16*x^2-76*x^3-21*x^4-2*x^5)/(1-x)^6) \\ G. C. Greubel, May 11 2017

From G. C. Greubel, May 11 2017: (Start)
G.f.: (1 - 6*x - 16*x^2 - 76*x^3 - 21*x^4 - 2*x^5)/(1 - x)^6.
E.g.f.: (1 - x - 15*x^2 - 25*x^3 - 10*x^4 - x^5)*exp(x). (End)

More terms from Harvey P. Dale, Feb 22 2016

A270390 Greatest common divisor of 2^n-1 and 5^n-1.

Original entry on oeis.org

1, 3, 1, 3, 1, 63, 1, 3, 1, 33, 1, 819, 1, 3, 31, 51, 1, 3591, 1, 1353, 1, 69, 1, 819, 1, 3, 1, 87, 1, 21483, 1, 51, 1, 3, 71, 1727271, 1, 3, 79, 1353, 1, 2408301, 1, 6141, 31, 141, 1, 13923, 1, 8283, 1, 159, 1, 10773, 1, 87, 1, 177, 1, 698476779, 1, 3, 1, 32691, 1

Offset: 1

Tom Edgar, Mar 16 2016

nonn

Ailon and Rudnick conjecture that a(n) = 1 infinitely often.

			For n=3, 2^3-1 = 7 and 5^3-1 = 124, thus a(3) = gcd(7,124) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
N. Ailon and Z. Rudnick, Torsion points on curves and common divisors of a^k-1 and b^k-1, arXiv:math/0202102 [math.NT], 2002; Acta Arith. 113 (2004), no. 1, 31-38.

Cf. A086892, A000225, A024049, A349748.

Maple
```
seq(igcd(2^n-1, 5^n-1), n=1..100);
```
Mathematica
```
Table[GCD[2^n - 1, 5^n - 1], {n, 100}]
```
PARI
```
vector(100,n,gcd(2^n-1,5^n-1))
```
Sage
```
[gcd(2^n-1,5^n-1) for n in [1..100]]
```

a(n) = gcd(2^n - 1, 5^n - 1).

a(n) = gcd(A000225(n), A024049(n)).

cp's OEIS Frontend

A074479 Largest prime factor of 5^n - 1.

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A366613 Sum of the divisors of 5^n-1.

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A024140 a(n) = 12^n - 1.

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A366612 Number of divisors of 5^n-1.

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A366611 Number of distinct prime divisors of 5^n - 1.

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A125831 a(n) = (5^n - 1)/2.

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A249433 Integers n such that n! does not divide the product of elements on row n of Pascal's triangle.

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