A004061 -id:A004061 - cp's OEIS frontend

A086123 a(n) = A004061(n) - 1.

Original entry on oeis.org

2, 6, 10, 12, 46, 126, 148, 180, 618, 928, 3406, 10948, 13240, 13872, 16518, 201358, 396412, 1888278, 3300592

Offset: 1

Labos Elemer, Jul 23 2003

Cf. A004061.

a(16)-a(19) added using the data at A004061 by Amiram Eldar, Jun 04 2024

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

Original entry on oeis.org

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)

A128344 Numbers k such that (7^k - 5^k)/2 is prime.

Original entry on oeis.org

3, 5, 7, 113, 397, 577, 7573, 14561, 58543, 100019, 123407, 136559, 208283, 210761, 457871, 608347, 636043

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.
No other terms less than 10^5. - Robert Price, May 28 2012
No other terms less than 10^6. - Jon Grantham, Jul 29 2023

Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A062572, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354. Cf. A004061, A082182, A121877, A059802. Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.

k=7; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((7^n-5^n)/2) \\ Charles R Greathouse IV, Feb 17 2017

a(7)-a(9) from Robert Price, May 28 2012

a(10)-a(17) from Jon Grantham, Jul 29 2023

A204940 Numbers n such that (23^n - 1)/22 is prime.

Original entry on oeis.org

5, 3181, 61441, 91943, 121949, 221411

Offset: 1

Robert Price, Jan 20 2012

No other terms < 100000.

H. Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A128000, A098438, A128002, A128003, A128004, A128005.

Select[Prime[Range[100]], PrimeQ[(23^#-1)/22]&]

is(n)=ispseudoprime((23^n-1)/22) \\ Charles R Greathouse IV, Jun 13 2017

a(5)=121949 corresponds to a probable prime discovered by Paul Bourdelais, Oct 19 2017

a(6)=221411 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A128336 Numbers k such that (6^k + 5^k)/11 is prime.

Original entry on oeis.org

3, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

No other terms less than 100000. - Robert Price, May 11 2012

Cf. A057171, A082387, A122853, A128335, A128337, A128338, A128339, A128340, A128341, A128342, A128343, A004061, A082182, A121877, A059802, A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=6; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

forprime(p=3,1e4,if(ispseudoprime((6^p+5^p)/11),print1(p", "))) \\ Charles R Greathouse IV, Jul 16 2011

a(7)-a(9) from Alexander Adamchuk, May 04 2010

One more term (8783) added (unknown discoverer) corresponding to a probable prime with 6834 digits by Jean-Louis Charton, Oct 06 2010

A128347 Numbers k such that (11^k - 5^k)/6 is prime.

Original entry on oeis.org

5, 41, 149, 229, 263, 739, 3457, 20269, 98221

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(10) > 10^5. - Robert Price, Jan 24 2013

Cf. A062572, A128344, A128345, A128346, A128348, A128349, A128350, A128351, A128352, A128353, A128354. Cf. A004061, A082182, A121877, A059802. Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.

k=11; Do[p=Prime[n]; f=(k^p-5^p)/(k-5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((11^n-5^n)/6) \\ Charles R Greathouse IV, Feb 17 2017

a(7)-a(9) from Robert Price, Jan 24 2013

A128342 Numbers k such that (13^k + 5^k)/18 is prime.

Original entry on oeis.org

13, 19, 31, 359, 487, 757, 761, 1667, 2551, 3167, 6829

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.
No other terms below 23600. - Max Alekseyev, Feb 01 2010
a(12) > 10^5. - Robert Price, Apr 30 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128343. Cf. A004061, A082182, A121877, A059802. Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=13; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((13^n+5^n)/18) \\ Charles R Greathouse IV, Feb 17 2017

Four more terms from Max Alekseyev, Feb 01 2010

A128341 Numbers k such that (12^k + 5^k)/17 is prime.

Original entry on oeis.org

3, 5, 13, 347, 977, 1091, 4861, 4967, 34679

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(10) > 10^5. - Robert Price, May 05 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128342, A128343.
Cf. A004061, A082182, A121877, A059802.
Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=12; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]
Select[Range[1100],PrimeQ[(12^#+5^#)/17]&] (* Harvey P. Dale, Jul 24 2012 *)

is(n)=isprime((12^n+5^n)/17) \\ Charles R Greathouse IV, Feb 17 2017

Two more terms (a(7) and a(8)) from Harvey P. Dale, Jul 24 2012

a(9) from Robert Price, May 05 2013

A128339 Numbers k such that (9^k + 5^k)/14 is prime.

Original entry on oeis.org

3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(12) > 10^5. - Robert Price, Dec 26 2012

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128340, A128341, A128342, A128343. Cf. A004061, A082182, A121877, A059802. Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

[n: n in [3..300] |IsPrime((9^n + 5^n) div 14)]; // Vincenzo Librandi, Nov 02 2018

k=9; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((9^n+5^n)/14) \\ Charles R Greathouse IV, Feb 17 2017

3 more PRP terms from Sean A. Irvine, Oct 01 2009

A128340 Numbers k such that (11^k + 5^k)/16 is prime.

Original entry on oeis.org

7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(11) > 10^5. - Robert Price, Feb 09 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128341, A128342, A128343.
Cf. A004061, A082182, A121877, A059802.
Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=11; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((11^n+5^n)/16) \\ Charles R Greathouse IV, Feb 17 2017

a(5)-a(10) from Robert Price, Feb 09 2013

cp's OEIS Frontend

A086123 a(n) = A004061(n) - 1.

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

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A128344 Numbers k such that (7^k - 5^k)/2 is prime.

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A204940 Numbers n such that (23^n - 1)/22 is prime.

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A128336 Numbers k such that (6^k + 5^k)/11 is prime.

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A128347 Numbers k such that (11^k - 5^k)/6 is prime.

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A128342 Numbers k such that (13^k + 5^k)/18 is prime.

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A128341 Numbers k such that (12^k + 5^k)/17 is prime.

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