A005058 -id:A005058 - cp's OEIS frontend

A109347 Zsigmondy numbers for a = 5, b = 3: Zs(n, 5, 3) is the greatest divisor of 5^n - 3^n (A005058) that is relatively prime to 5^m - 3^m for all positive integers m < n.

2, 1, 49, 17, 1441, 19, 37969, 353, 19729, 421, 24325489, 481, 609554401, 10039, 216001, 198593, 381405156481, 12979, 9536162033329, 288961, 18306583, 6125659, 5960417405949649, 346561, 103408180634401, 152787181, 3853528045489, 179655841, 93132223146359169121

Offset: 1

Jonathan Vos Post, Aug 21 2005

nonn

Eric Weisstein's World of Mathematics, Zsigmondy's Theorem

Cf. A064078, A064079, A064080, A064081, A064082, A064083, A109325, A109348, A109349.

rad(n) = factorback(factor(n)[, 1])
lista(nn) = {prad = 1; for (n=1, nn, val = 5^n-3^n; d = divisors(val); gd = 1; forstep(k=#d, 1, -1, if (gcd(d[k], prad) == 1, g = d[k]; break)); print1(g, ", "); prad = ra(prad*val););} \\ Michel Marcus, Nov 15 2016

Edited, corrected and extended by Ray Chandler, Aug 26 2005
Definition corrected by Jerry Metzger, Nov 04 2009
More terms from Michel Marcus, Nov 14 2016

A121877 Numbers k such that (5^k - 3^k)/2 = A005059(k) is prime.

Original entry on oeis.org

13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, 128941, 147571, 182099, 866029

Offset: 1

Alexander Adamchuk, Aug 31 2006, Oct 08 2006

All terms are primes. Their indices are listed in A123704.
Corresponding primes are listed in A123705.
If it exists, a(17) > 125000. - Robert Price, Aug 15 2011
If it exists, a(21) > 1000000. - Jon Grantham, Jul 29 2023

OEIS Wiki, Primes of the form (a^n+b^n)/(a+b) and (a^n-b^n)/(a-b).
Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.

Cf. A005058, A005059, A109347, A120612, A081186, A121824, A123704, A123705.

Do[f=(5^n-3^n)/2;If[PrimeQ[f],Print[{n,f}]],{n,1,300}]

forprime(p=2,1e4,if(ispseudoprime((5^p-3^p)>>1),print1(p", "))) \\ Charles R Greathouse IV, Jun 16 2011

a(n) = prime(A123704(n)).

More terms from Farideh Firoozbakht, Oct 11 2006
a(13)-a(16) from Robert Price, Aug 15 2011
a(17)-a(19) from Kellen Shenton, May 18 2022
a(20) from Jon Grantham, Jul 29 2023

A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

Original entry on oeis.org

3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789

Offset: 1

Alexander Adamchuk, Sep 14 2006

(3^k + 5^k)/8 = A074606(k)/8 = A081186(k)/4.
Corresponding primes of the form (3^k + 5^k)/2^3 are listed in {A121938(n)} = {A079773(a(n))} = {19, 421, 10039, 95383574161, 2384331073699, ...}.
No other terms less than 100000. - Robert Price, Apr 28 2012

Cf. A074606, A081186, A121824, A121877, A005058, A005059, A121938, A109347, A079773.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}]

select(n->isprime((3^n + 5^n)/8), vector(2000,i,i)) \\ Charles R Greathouse IV, Feb 13 2011

a(11)-a(15) from Robert Price, Apr 28 2012

A007798 Expected number of random moves in Tower of Hanoi problem with n disks starting with a randomly chosen position and ending at a position with all disks on the same peg.

Original entry on oeis.org

0, 0, 2, 18, 116, 660, 3542, 18438, 94376, 478440, 2411882, 12118458, 60769436, 304378620, 1523487422, 7622220078, 38125449296, 190670293200, 953480606162, 4767790451298, 23840114517956, 119204059374180, 596030757224102, 2980185167180118, 14901019979079416

Offset: 0

David G. Poole (dpoole(AT)trentu.ca)

All 3^n possible starting positions are chosen with equal probability.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. A. Alekseyev and T. Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).
Index entries for sequences related to Towers of Hanoi

Partial sums of A005058.

Cf. A134939.

[(5^n-2*3^n+1)/4: n in [0..25]]; // Vincenzo Librandi, Oct 11 2011

seq( (1 -2*3^n +5^n)/4, n=0..25); # G. C. Greubel, Mar 05 2020

Table[(1 -2*3^n +5^n)/4, {n,0,25}] (* G. C. Greubel, Mar 05 2020 *)

concat([0,0], Vec(-2*x^2/((x-1)*(3*x-1)*(5*x-1)) + O(x^30))) \\ Colin Barker, Sep 17 2014

[(1 -2*3^n +5^n)/4 for n in (0..25)] # G. C. Greubel, Mar 05 2020

For n>1, a(n) = 8*a(n-1) - 15*a(n-2) + 2 = 2*A016209(n-2). - Henry Bottomley, Jun 06 2000
a(n) = (5^n - 2*3^n + 1) / 4. - Henry Bottomley, Jun 06 2000, proved by Max Alekseyev, Feb 04 2008
From Colin Barker, Sep 17 2014: (Start)
a(n) = 9*a(n-1) - 23*a(n-2) + 15*a(n-3).
G.f.: 2*x^2/((1-x)*(1-3*x)*(1-5*x)). (End)
E.g.f.: (exp(x) - 2*exp(3*x) + exp(5*x))/4. - G. C. Greubel, Mar 05 2020

More precise definition and more terms from Max Alekseyev, Feb 04 2008

a(0)=0 prepended by Max Alekseyev, Sep 08 2014

A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

Original entry on oeis.org

19, 421, 10039, 95383574161, 2384331073699, 1925929944387235853055979210606894889560480247048440342330377620014353281101

Offset: 1

Zak Seidov, Sep 10 2006

Corresponding numbers k such that (3^k + 5^k)/8 is prime are listed in A122853. All these numbers are primes. - Alexander Adamchuk, Sep 14 2006

The next term is too large to include. - Alexander Adamchuk, Sep 14 2006

Cf. A074706, A122853, A081186, A079773, A121824, A121877, A005058, A005059, A121938, A109347.

Do[f=5^n+3^n;If[PrimeQ[f/2^3],Print[{n,f/2^3}]],{n,1,1231}] (* Alexander Adamchuk, Sep 14 2006 *)

a(n) = (A122853(n)^3 + A122853(n)^5)/8. a(n) = A074606[A122853(n)]/8 = A081186[A122853(n)]/4. a(n) = A079773[A122853(n)]. - Alexander Adamchuk, Sep 14 2006

More terms from Alexander Adamchuk, Sep 14 2006

A195986 Exponent of the largest power of 2 that divides 5^n - 3^n.

Original entry on oeis.org

1, 4, 1, 5, 1, 4, 1, 6, 1, 4, 1, 5, 1, 4, 1, 7, 1, 4, 1, 5, 1, 4, 1, 6, 1, 4, 1, 5, 1, 4, 1, 8, 1, 4, 1, 5, 1, 4, 1, 6, 1, 4, 1, 5, 1, 4, 1, 7, 1, 4, 1, 5, 1, 4, 1, 6, 1, 4, 1, 5, 1, 4, 1, 9, 1, 4, 1, 5, 1, 4, 1, 6, 1, 4, 1, 5, 1, 4, 1, 7, 1, 4, 1, 5, 1, 4

Offset: 1

John W. Layman, Oct 12 2011

nonn

Conjecture: a(n) = 1 if A090740 = 1, else a(n) = A090740(n)+1.

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A005058, A007814, A090740.

Table[IntegerExponent[5^n - 3^n, 2], {n, 100}] (* T. D. Noe, Oct 12 2011 *)

A195986(n) = valuation(5^n - 3^n,2); \\ Antti Karttunen, Nov 06 2018

a(n) = A007814(A005058(n)). - Antti Karttunen, Nov 06 2018

Name clarified by Antti Karttunen, Nov 06 2018

A226511 a(n) = 3*(5^n-3^n)/2.

Original entry on oeis.org

0, 3, 24, 147, 816, 4323, 22344, 113907, 576096, 2900163, 14559864, 72976467, 365413776, 1828663203, 9148098984, 45754843827, 228817265856, 1144215469443, 5721464767704, 28608486099987, 143045917284336, 715240046774883, 3576231614934024, 17881252217848947, 89406543518781216

Offset: 0

N. J. A. Sloane, Jun 12 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Max A. Alekseyev and T. Berger, Solving the Tower of Hanoi with Random Moves. In: J. Beineke, J. Rosenhouse (eds.) The Mathematics of Various Entertaining Subjects: Research in Recreational Math, Princeton University Press, 2016, pp. 65-79. ISBN 978-0-691-16403-8
Index entries for linear recurrences with constant coefficients, signature (8,-15).

Cf. A005058.

[(3/2)*(5^n-3^n): n in [0..30]]; // Vincenzo Librandi, Jun 12 2013

CoefficientList[Series[3 x / ((1 - 5 x) (1 - 3 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 12 2013 *)
Table[3 (5^n - 3^n)/2, {n, 0, 25}] (* Bruno Berselli, Jun 12 2013 *)

G.f.: 3*x/((1-5*x)*(1-3*x)). - Vincenzo Librandi, Jun 12 2013
a(n) = 8*a(n-1) -15*a(n-2) for n>1, a(0)=0, a(1)=3. - Vincenzo Librandi, Jun 12 2013
a(n) = A005059(n)*3 = A005058(n)*3/2.

A248225 a(n) = 6^n - 3^n.

Original entry on oeis.org

0, 3, 27, 189, 1215, 7533, 45927, 277749, 1673055, 10058013, 60407127, 362619909, 2176250895, 13059099693, 78359381127, 470170635669, 2821066860735, 16926530304573, 101559569247927, 609358577749029, 3656154953278575, 21936940180024653

Offset: 0

Vincenzo Librandi, Oct 04 2014

Index entries for linear recurrences with constant coefficients, signature (9,-18).

Cf. sequences of the form k^n-3^n: A005061 (k=4), A005058 (k=5), this sequence (k=6), A190541 (k=7), A190543 (k=8), A059410 (k=9), A248226 (k=10), A139741 (k=11).

Cf. A000225, A000244.

Magma
```
[6^n-3^n: n in [0..30]];
```

Table[6^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[3 x / ((1 - 3 x) (1 - 6 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-18},{0,3},30] (* Harvey P. Dale, Jul 12 2025 *)

G.f.: 3*x/((1-3*x)*(1-6*x)).
a(n) = 9*a(n-1) - 18*a(n-2).
a(n) = 3^n*(2^n - 1) = A000244(n)*A000225(n).
E.g.f.: 2*exp(9*x/2)*sinh(3*x/2). - Elmo R. Oliveira, Mar 31 2025

A120948 8n+3^n+5^n.

Original entry on oeis.org

2, 16, 50, 176, 738, 3408, 16402, 80368, 397250, 1972880, 9824754, 49005360, 244672162, 1222297552, 6108298706, 30531927152, 152630937474, 763068593424, 3815084686258, 19074648589744, 95370918425186, 476847618556496

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[8*n+3^n+5^n: n in [0..25]]; // Bruno Berselli, Feb 27 2013

CoefficientList[Series[2 (1 - 2 x - 23 x^2 + 56 x^3)/((1-5 x) (1 - 3 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 26 2013 *)
Table[8 n + 3^n + 5^n, {n, 0, 25}] (* Bruno Berselli, Feb 27 2013 *)

for(n=0, 25, print1(8*n+3^n+5^n", ")); \\ Bruno Berselli, Feb 27 2013

G.f.: 2*(1-2*x-23*x^2+56*x^3)/((1-5*x)*(1-3*x)*(1-x)^2). - Vincenzo Librandi, Feb 26 2013

Edited by Ray Chandler, Sep 06 2006

A120949 2n+3^n+5^n.

Original entry on oeis.org

2, 10, 38, 158, 714, 3378, 16366, 80326, 397202, 1972826, 9824694, 49005294, 244672090, 1222297474, 6108298622, 30531927062, 152630937378, 763068593322, 3815084686150, 19074648589630, 95370918425066, 476847618556370

Offset: 0

Mohammad K. Azarian, Aug 19 2006

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-32,38,-15).

Cf. A074606, A005058.

[2*n+3^n+5^n: n in [0..30]]; // Vincenzo Librandi, Feb 27 2013

Table[2 n + 3^n + 5^n, {n, 0, 30}] (* or *) CoefficientList[Series[2 (1 - 5 x + x^2 + 11 x^3)/((1 -5 x) (1 - 3 x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 26 2013 *)
LinearRecurrence[{10,-32,38,-15},{2,10,38,158},30] (* Harvey P. Dale, Jan 18 2021 *)

a(n)=2*n+3^n+5^n \\ Charles R Greathouse IV, Feb 27 2013

G.f.: 2*(1-5*x+x^2+11*x^3)/((1-5*x)*(1-3*x)*(1-x)^2). - Vincenzo Librandi, Feb 26 2013

Edited by Ray Chandler, Sep 06 2006

cp's OEIS Frontend

A109347 Zsigmondy numbers for a = 5, b = 3: Zs(n, 5, 3) is the greatest divisor of 5^n - 3^n (A005058) that is relatively prime to 5^m - 3^m for all positive integers m < n.

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A121877 Numbers k such that (5^k - 3^k)/2 = A005059(k) is prime.

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A122853 Numbers k such that (3^k + 5^k)/8 = A074606(k)/8 is a prime.

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A007798 Expected number of random moves in Tower of Hanoi problem with n disks starting with a randomly chosen position and ending at a position with all disks on the same peg.

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A121938 Primes of the form (3^k + 5^k)/2^3 = A074606(k)/8.

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A195986 Exponent of the largest power of 2 that divides 5^n - 3^n.

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PARI

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A226511 a(n) = 3*(5^n-3^n)/2.

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Magma

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A248225 a(n) = 6^n - 3^n.