A074479 -id:A074479 - cp's OEIS frontend

A003463 a(n) = (5^n - 1)/4.

0, 1, 6, 31, 156, 781, 3906, 19531, 97656, 488281, 2441406, 12207031, 61035156, 305175781, 1525878906, 7629394531, 38146972656, 190734863281, 953674316406, 4768371582031, 23841857910156, 119209289550781, 596046447753906, 2980232238769531

Offset: 0

N. J. A. Sloane

5^a(n) is the highest power of 5 dividing (5^n)!. - Benoit Cloitre, Feb 04 2002
n such that A002294(n) is not divisible by 5. - Benoit Cloitre, Jan 14 2003
Without leading zero, i.e., sequence {a(n+1) = (5*5^n-1)/4}, this is the binomial transform of A003947. - Paul Barry, May 19 2003 [Edited by M. F. Hasler, Oct 31 2014]
Numbers n such that a(n) is prime are listed in A004061(n) = {3, 7, 11, 13, 47, 127, 149, 181, 619, 929, ...}. Corresponding primes a(n) are listed in A086122(n) = {31, 19531, 12207031, 305175781, 177635683940025046467781066894531, ...}. 3^(m+1) divides a(2*3^m*k). 31 divides a(3k). 13 divides a(4k). 11 divides a(5k). 71 divides a(5k). 7 divides a(6k). 19531 divides a(7k). 313 divides a(8k). 19 divides a(9k). 829 divides a(9k). 71 divides a(10k). 521 divides a(10k). 17 divides a(16k). p divides a(p-1) for all prime p except p = {2,5}. p^(m+1) divides a(p^m*(p-1)) for all prime p except p = {2,5}. p divides a((p-1)/2) for prime p = {11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, 109, ...} = A045468, Primes congruent to {1, 4} mod 5. p divides a((p-1)/3) for prime p = {13, 67, 127, 163, 181, 199, 211, 241, 313, 337, 367, 379, 409, 457, ...}. p divides a((p-1)/4) for prime p = {101, 109, 149, 181, 269, 389, 401, 409, 449, 461, 521, 541, ...} = A107219, Primes of the form x^2+100y^2. p divides a((p-1)/5) for prime p = {31, 191, 251, 271, 601, 641, 761, 1091, 1861, ...}. p divides a((p-1)/6) for prime p = {181, 199, 211, 241, 379, 409, 631, 691, 739, 769, 1039, ...}. - Alexander Adamchuk, Jan 23 2007
Starting with 1 = convolution square of A026375: (1, 3, 11, 45, 195, 873, ...). - Gary W. Adamson, May 17 2009
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=5, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 27 2010
This is the sequence A(0,1;4,5;2) = A(0,1;6,-5;0) of the family of sequences [a,b:c,d:k] considered by Gary Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010
It is the Lucas sequence U(6,5). - Felix P. Muga II, Mar 21 2014
a(2*n+1) is the sum of the numerators and denominators of the reduced fractions 0 < b/5^n < 1 plus 1, with b < 5^n. - J. M. Bergot, Jul 24 2015
The sequence multiplied by 10 (0, 10, 60, 310, 1560, ...) is the maximum number of coins which can be decided by n weighings on 2 balances in the counterfeit coin problem with undecided under/overweight. [Halbeisen and Hungerbuhler, Disc. Math. 147 (1995) 139 Theorem 1]. - R. J. Mathar, Sep 10 2015
Order of the rank-n projective geometry PG(n-1,5) over the finite field GF(5). - Anthony Hernandez, Oct 05 2016
Number of zeros in the substitution system {0 -> 11100, 1 -> 11110} at step n from initial string "1" (1 -> 11110 -> 1111011110111101111011100 -> ...). - Ilya Gutkovskiy, Apr 10 2017
a(n) is the numerator of Sum_{k=1..n} 1/5^k, which approaches a limit of 1/4. The denominators are 5^n. In general, Sum_{k=1..n} 1/x^k approaches a limit of 1/(x-1). It is of interest to note that as x increases, so does the rate of convergence. See Crossrefs for numerators for other values of x which have the general form (x^n-1)/(x-1). - Gary Detlefs, Aug 31 2021

			Base 5...........decimal
0......................0
1......................1
11.....................6
111...................31
1111.................156
11111................781
111111..............3906
1111111............19531
11111111...........97656
111111111.........488281
1111111111.......2441406
etc. ...............etc.
- _Zerinvary Lajos_, Jan 14 2007

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 282.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joseph E. Bonin and Joseph P. S. Kung The Number of Points In A Combinatorial Geometry With No 8-Point-Line Minors, Mathematical Essays in Honor of Gian-Carlo Rota, B. Sagan and R. P. Stanley, eds., Birkhäuser, 1998, 271-284.
Carlos M. da Fonseca and Anthony G. Shannon, A formal operator involving Fermatian numbers, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 491-498.
Roger B. Eggleton, Maximal Midpoint-Free Subsets of Integers, International Journal of Combinatorics Volume 2015, Article ID 216475, 14 pages.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 374
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Eric Weisstein's World of Mathematics, Repunit
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A004061, A026375, A045468, A060458, A074479, A086122, A107219.

Cf. A003462, A002450, A003464, A023000, A023001, A002452, A002275.

[(5^n-1)/4 : n in [0..30]]; // Wesley Ivan Hurt, Sep 25 2014

a:=n->sum(5^(n-j),j=1..n): seq(a(n), n=0..23); # Zerinvary Lajos, Jan 04 2007
A003463:=1/(5*z-1)/(z-1); # Simon Plouffe in his 1992 dissertation
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=6*a[n-1]-5*a[n-2]od: seq(a[n], n=0..23); # Zerinvary Lajos, Feb 21 2008

lst={}; Do[p=(5^n-1)/4; AppendTo[lst, p], {n, 0, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 29 2008 *)
Table[((5^n-1)/4),{n,0,25}] (* Vincenzo Librandi, Aug 20 2012 *)
NestList[5 # + 1 &, 0, 23] (* Bruno Berselli, Feb 06 2013 *)
LinearRecurrence[{6,-5},{0,1},30] (* Harvey P. Dale, Sep 20 2023 *)

A003463(n):=floor((5^n-1)/4)$ makelist(A003463(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=5^n\4; \\ Charles R Greathouse IV, Jul 15 2011

[lucas_number1(n,6,5) for n in range(0, 24)] # Zerinvary Lajos, Apr 22 2009

[gaussian_binomial(n,1,5) for n in range(0,24)] # Zerinvary Lajos, May 28 2009

Second binomial transform of A015518; binomial transform of A000302 (preceded by 0). - Paul Barry, Mar 28 2003
a(n) = Sum_{k=1..n} binomial(n,k)*4^(k-1). - Paul Barry, Mar 28 2003
a(n) = (-1)^n times the (i, j)-th element of M^n (for all i and j such that i is not equal to j), where M = ((1, -1, 1, -2), (-1, 1, -2, 1), (1, -2, 1, -1), (-2, 1, -1, 1)). - Simone Severini, Nov 25 2004
a(n) = A125118(n,4) for n>3. - Reinhard Zumkeller, Nov 21 2006
a(n) = ((3+sqrt(4))^n - (3-sqrt(4))^n)/4. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008
a(n) = 6*a(n-1) - 5*a(n-2) n>1, a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
From Wolfdieter Lang, Oct 18 2010: (Start)
O.g.f.: x/((1-5*x)*(1-x)).
a(n) = 4*a(n-1) + 5*a(n-2) + 2, a(0)=0, a(1)=1.
a(n) = 5*a(n-1) + a(n-2) - 5*a(n-3) = 7*a(n-1) - 11*a(n-2) + 5*a(n-3), a(0)=0, a(1)=1, a(2)=6. Observation by G. Detlefs. See the W. Lang comment and link. (End)
a(n) = 5*a(n-1) + 1 with n>0, a(0)=0. - Vincenzo Librandi, Nov 17 2010
a(n) = a(n-1) + A000351(n-1) n>0, a(0)=0. - Felix P. Muga II, Mar 19 2014
a(n) = a(n-1) + 20*a(n-2) + 5 for n > 1, a(0)=0, a(1)=1. - Felix P. Muga II, Mar 19 2014
a(n) = A060458(n)/2^(n+2), for n > 0. - R. J. Cano, Sep 25 2014
From Ilya Gutkovskiy, Oct 05 2016: (Start)
E.g.f.: (exp(4*x) - 1)*exp(x)/4.
Convolution of A000351 and A057427. (End)

A057956 Number of prime factors of 5^n - 1 (counted with multiplicity).

Original entry on oeis.org

2, 4, 3, 6, 4, 7, 3, 8, 5, 7, 3, 10, 3, 7, 7, 11, 4, 11, 5, 11, 6, 8, 4, 13, 8, 7, 9, 10, 5, 14, 4, 14, 6, 8, 9, 16, 5, 10, 6, 15, 4, 16, 4, 12, 12, 8, 3, 17, 4, 13, 8, 12, 5, 19, 10, 13, 7, 9, 4, 21, 5, 9, 11, 18, 8, 15, 7, 14, 9, 16, 4, 22, 5, 10, 16, 14, 7, 14, 5, 20, 11, 10, 5, 22, 9, 10

Offset: 1

Patrick De Geest, Nov 15 2000

nonn

Max Alekseyev, Table of n, a(n) for n = 1..502 (first 448 terms from Amiram Eldar)
S. S. Wagstaff, Jr., The Cunningham Project

bigomega(b^n-1): A057951 (b=10), A057952 (b=9), A057953 (b=8), A057954 (b=7), A057955 (b=6), this sequence (b=5), A057957 (b=4), A057958 (b=3), A046051 (b=2).

Cf. A001222, A024049, A074479, A085030.

PrimeOmega[5^Range[90]-1] (* Harvey P. Dale, Dec 16 2017 *)

Mobius transform of A085030. - T. D. Noe, Jun 19 2003

a(n) = A001222(A024049(n)). - Amiram Eldar, Feb 01 2020

A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

Original entry on oeis.org

3, 5, 3, 12, 9, 37, 3, 28, 18, 47, 3, 180, 3, 53, 81, 176, 9, 446, 21, 564, 39, 117, 9, 884, 180, 53, 360, 244, 21, 5959, 9, 800, 39, 111, 369, 9536, 21, 483, 39, 5476, 9, 18289, 9, 1140, 2958, 111, 3, 9424, 6, 3852, 177, 884, 21, 81048, 561, 1188, 69, 227, 9

Offset: 1

Vladeta Jovovic, Feb 06 2001

nonn

The multiplicative order of a mod m, gcd(a,m)=1, is the smallest natural number d for which a^d = 1 (mod m). a(n) = number of orders of degree-n monic irreducible polynomials over GF(5).

Also, number of primitive factors of 5^n - 1 (cf. A218357). - Max Alekseyev, May 03 2022

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

Number of primitive factors of b^n - 1: A059499 (b=2), A059885(b=3), A059886 (b=4), this sequence (b=5), A059888 (b=6), A059889 (b=7), A059890 (b=8), A059891 (b=9), A059892 (b=10).
Cf. A000005, A008683, A001692, A053447, A057956, A058945, A074479, A143665, A212485, A218357
Column k=5 of A212957.

with(numtheory):
a:= n-> add(mobius(n/d)*tau(5^d-1), d=divisors(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 12 2012

a[n_] := Sum[MoebiusMu[n/d]*DivisorSigma[0, 5^d-1], {d, Divisors[n]}];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Dec 13 2024, after Alois P. Heinz *)

a(n) = sumdiv(n, d, moebius(n/d)*numdiv(5^d-1)); \\ Michel Marcus, Dec 13 2024

a(n) = Sum_{d|n} mu(n/d)*tau(5^d-1), (mu(n) = Moebius function A008683, tau(n) = number of divisors of n A000005).

A274906 Largest prime factor of 4^n - 1.

Original entry on oeis.org

3, 5, 7, 17, 31, 13, 127, 257, 73, 41, 683, 241, 8191, 127, 331, 65537, 131071, 109, 524287, 61681, 5419, 2113, 2796203, 673, 4051, 8191, 262657, 15790321, 3033169, 1321, 2147483647, 6700417, 599479, 131071, 122921, 38737, 616318177, 525313, 22366891

Offset: 1

Vincenzo Librandi, Jul 11 2016

nonn

			4^7 - 1 = 16383 = 3*43*127, so a(7) = 127

Max Alekseyev, Table of n, a(n) for n = 1..1128
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Second bisection of A005420. - Michel Marcus, Jul 13 2016
Cf. largest prime factor of k^n-1: A005420 (k=2), A074477 (k=3), this sequence (k=4), A074479 (k=5), A274907 (k=6), A074249 (k=7), A274908 (k=8), A274909 (k=9), A005422 (k=10), A274910 (k=11).
Cf. A006530, A024036.

[Maximum(PrimeDivisors(4^n-1)): n in [1..40]];

Table[FactorInteger[4^n - 1][[-1, 1]], {n, 40}]

a(n) = A006530(A024036(n)). - Michel Marcus, Jul 11 2016

a(n) = max(A002587(n),A005420(n)). - Max Alekseyev, Apr 25 2022

Terms to a(100) in b-file from Vincenzo Librandi, Jul 13 2016
a(101)-a(603) in b-file from Amiram Eldar, Feb 08 2020
a(604)-a(1128) in b-file from Max Alekseyev, Jul 25 2023, Mar 15 2025

A074477 Largest prime factor of 3^n - 1.

Original entry on oeis.org

2, 2, 13, 5, 11, 13, 1093, 41, 757, 61, 3851, 73, 797161, 1093, 4561, 193, 34511, 757, 363889, 1181, 368089, 3851, 1001523179, 6481, 391151, 797161, 8209, 16493, 20381027, 4561, 4404047, 21523361, 2413941289, 34511, 2664097031, 530713

Offset: 1

Rick L. Shepherd, Aug 23 2002

nonn

			3^7 - 1 = 2186 = 2*1093, so a(7) = 1093.

Table of n, a(n) for n = 1..690
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A006530 (largest prime factor), A024023 (3^n-1).

Cf. A074476 (largest prime factor of 3^n + 1), A005420 (largest prime factor of 2^n - 1), A074479 (largest prime factor of 5^n - 1).

[Maximum(PrimeDivisors(3^n-1)): n in [1..40]]; // Vincenzo Librandi, Aug 23 2013

A074477 := proc(n)
        A006530( 3^n-1) ;
end proc: # R. J. Mathar, Jul 18 2015
# alternative:
a:= n-> max(seq(i[1], i=ifactors(3^n-1)[2])):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 18 2015

Table[FactorInteger[3^n - 1] [[-1, 1]], {n, 40}] (* Vincenzo Librandi, Aug 23 2013 *)

for(n=1,40, v=factor(3^n-1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A024023(n)). - Michel Marcus, Jul 18 2015

Terms to a(100) in b-file from Vincenzo Librandi, Aug 23 2013
a(101)-a(660) in b-file from Amiram Eldar, Feb 01 2020
a(661)-a(690) in b-file from Max Alekseyev, May 22 2022

A074478 Largest prime factor of 5^n + 1.

Original entry on oeis.org

2, 3, 13, 7, 313, 521, 601, 449, 11489, 5167, 9161, 5281, 390001, 38923, 234750601, 7621, 29423041, 41540861, 6597973, 213029, 632133361, 7603, 1030330938209, 42272797713043, 152587500001, 50150933101, 83181652304609, 16018507

Offset: 0

Rick L. Shepherd, Aug 23 2002

nonn

			5^11 + 1 = 48828126 = 2*3*23*67*5281, so a(11) = 5281.

Table of n, a(n) for n = 0..471
S. S. Wagstaff, Jr., The Cunningham Project

Cf. A002587 (largest prime factor of 2^n + 1), A074479 (largest prime factor of 5^n - 1), A074476 (largest prime factor of 3^n + 1), A227575 (largest prime factor of 7^n + 1).

Cf. A006530, A034474.

[Maximum(PrimeDivisors(5^n+1)): n in [0..30]]; // Vincenzo Librandi, Jul 09 2016

Table[FactorInteger[5^n + 1][[-1, 1]], {n, 0, 30}] (* Bruno Berselli, Aug 23 2013 *)

for(n=0,30, v=factor(5^n+1); print1(v[matsize(v)[1],1],","))

a(n) = A006530(A034474(n)). - Michel Marcus, Jul 09 2016

Terms to a(100) in b-file from Vincenzo Librandi, Jul 09 2016
a(101)-a(451) in b-file from Amiram Eldar, Feb 01 2020
a(452)-a(471) in b-file from Max Alekseyev, Apr 25 2022, Jan 04 2024

cp's OEIS Frontend

A003463 a(n) = (5^n - 1)/4.

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A057956 Number of prime factors of 5^n - 1 (counted with multiplicity).

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A059887 a(n) = |{m : multiplicative order of 5 mod m=n}|.

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A274906 Largest prime factor of 4^n - 1.

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A074477 Largest prime factor of 3^n - 1.

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A074478 Largest prime factor of 5^n + 1.

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