A086122 -id:A086122 - 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)

A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

Original entry on oeis.org

7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003

Primes that are base-b repunits with three or more digits for at least one b >= 2: Primes in A053696. Subsequence of A000668 U A076481 U A086122 U A165210 U A102170 U A004022 U ... (for each possible b). - Rick L. Shepherd, Sep 07 2009
From Bernard Schott, Dec 18 2012: (Start)
Also known as Brazilian primes. The primes that are not Brazilian primes are in A220627.
The number of terms k+1 is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3*37.
The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33 (see Theorem 4 of Quadrature article in the Links). (End)
It is not known whether there are infinitely many Brazilian primes. See A002383. - Bernard Schott, Jan 11 2013
Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - Thomas Ordowski, Apr 25 2013
Number of terms less than 10^n: 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ..., . - Robert G. Wilson v, Mar 31 2014
From Bernard Schott, Apr 08 2017: (Start)
Brazilian primes fall into two classes:
1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,
2) when n is composite, we get sequence A285017. (End)
The conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link Bernard Schott, Quadrature, Conjecture 1, page 36) is false. Thanks to Giovanni Resta, who found that a(856) = 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73 is the 141385th Sophie Germain prime. - _Bernard Schott, Mar 08 2019

			13 is a term since it is prime and 13 = 1 + 3 + 3^2 = 111_3.
31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = 11111_2.
From _Hartmut F. W. Hoft_, May 08 2017: (Start)
The sequence represented as a sparse matrix with the k-th column indexed by A006093(k+1), primes minus 1, and row n by A000027(n+1). Traversing the matrix by counterdiagonals produces a non-monotone ordering.
    2    4      6        10             12          16
2  7    31     127      -              8191        131071
3  13   -      1093     -              797161      -
4  -    -      -        -              -           -
5  31   -      19531    12207031       305175781   -
6  43   -      55987    -              -           -
7  -    2801   -        -              16148168401 -
8  73   -      -        -              -           -
9  -    -      -        -              -           -
10  -    -      -        -              -           -
11  -    -      -        -              -           50544702849929377
12  157  22621  -        -              -           -
13  -    30941  5229043  -              -           -
14  211  -      8108731  -              -           -
15  241  -      -        -              -           -
16 -    -      -        -              -           -
17  307  88741  25646167 2141993519227  -           -
18  -    -      -        -              -           -
19  -    -      -        -              -           -
20  421  -      -        10778947368421 -           689852631578947368421
21  463  -      -        17513875027111 -           1502097124754084594737
22  -    245411 -        -              -           -
23  -    292561 -        -              -           -
24  601  346201 -        -              -           -
Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:
column  2:  A002383            row 2:  A000668
column  4:  A088548            row 3:  A076481
column  6:  A088550            row 4:  -
column 10:  A162861            row 5:  A086122.
(End)

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10831 (terms up to 10^10; terms 1..3880 from T. D. Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A189891 (complement), A125134 (Brazilian numbers), A220627 (Primes that are non-Brazilian).
Cf. A003424 (n restricted to prime powers).
Cf. A053696, A086930, A059055.
Equals A023195 \3 Union A285017 with empty intersection.
Primes of the form (b^k-1)/(b-1) for b=2: A000668, b=3: A076481, b=5: A086122, b=6: A165210, b=7: A102170, b=10: A004022.
Primes of the form (b^k-1)/(b-1) for k=3: A002383, k=5: A088548, k=7: A088550, k=11: A162861.

a085104 n = a085104_list !! (n-1)
a085104_list = filter ((> 1) . a088323) a000040_list
-- Reinhard Zumkeller, Jan 22 2014

max = 140; maxdata = (1 - max^3)/(1 - max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 - m^i)/(1 - m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)
f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[ id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Select[ Range[2, 60], 1 + f@# != Prime@# &] (* Robert G. Wilson v, Mar 31 2014 *)

list(lim)=my(v=List(),t,k);for(n=2,sqrt(lim), t=1+n;k=1; while((t+=n^k++)<=lim,if(isprime(t), listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 08 2013

A085104_vec(N,L=List())=forprime(K=3,logint(N+1,2),for(n=2,sqrtnint(N-1,K-1),isprime((n^K-1)\(n-1))&&listput(L,(n^K-1)\(n-1))));Set(L) \\ M. F. Hasler, Jun 26 2018

A010051(a(n)) * A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

More terms from David Wasserman, Jan 26 2005

A165210 Primes of the form (6^m - 1)/5.

Original entry on oeis.org

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371

Offset: 1

Rick L. Shepherd, Sep 07 2009

nonn

Prime repunits in base 6 whose representation consists of m 1's. The exponents m are in A004062. a(5) and a(6) have 55 and 99 decimal digits, respectively.

			a(2) = (6^A004062(2) - 1)/5 = (6^3 - 1)/5 = 215/5 = 43, which is 111_6.

Vincenzo Librandi, Table of n, a(n) for n = 1..9

Cf. A004062, A003464, A085104, A000668, A076481, A086122, A102170, A004022.

[a: n in [1..200] | IsPrime(a) where a is  (6^n-1) div 5 ]; // Vincenzo Librandi, Dec 09 2011

Select[Table[(6^n-1)/5, {n,0,2000}], PrimeQ] (* Vincenzo Librandi, Dec 09 2011 *)

a(n) = (6^A004062(n) - 1)/5.

A193574 Smallest divisor of sigma(n) that does not divide n.

Original entry on oeis.org

3, 2, 7, 2, 4, 2, 3, 13, 3, 2, 7, 2, 3, 2, 31, 2, 13, 2, 3, 2, 3, 2, 5, 31, 3, 2, 8, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 31, 3, 3, 2, 7, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 127, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3

Offset: 2

Franklin T. Adams-Watters, Aug 27 2011

nonn

a(n) = 2 iff n is an odd number that is not a perfect square.
From Hartmut F. W. Hoft, May 05 2017: (Start)
(1)  Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
(2)  n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v  with  u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
(3)  The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
(4)  Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
(5)  Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p)
(6)  Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime.
(End)
Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - David A. Corneth, May 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000203, A007978, A027750, A135718.

Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.

import Data.List ((\\))
a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
   where nDivisors = a027750_row n
         sigma = sum nDivisors
-- Reinhard Zumkeller, May 20 2015, Aug 28 2011

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

a(n)=local(ds);ds=divisors(sigma(n));for(k=2,#ds,if(n%ds[k],return(ds[k])))

cp's OEIS Frontend

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

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A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

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A165210 Primes of the form (6^m - 1)/5.

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A193574 Smallest divisor of sigma(n) that does not divide n.

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A086124 Primes generated by linear recursion: f(n) = prime(n) * f(n-1) + 2, f(1) = 1.

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A086125 Values of k such that f(k) is a prime, where f(1) = 1, f(i) = prime(i)*f(i-1) + 2.

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A179625 Legal generalized repunit prime numbers.

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