A059613 -id:A059613 - cp's OEIS frontend

A124621 Numbers k such that 5^k + 4 is prime.

0, 2, 6, 10, 102, 494, 794, 1326, 5242, 5446, 24602, 87606, 188558

Offset: 1

Artur Jasinski, Dec 21 2006

nonn

a(13) > 10^5. - Robert Price, Feb 03 2014

a(14) > 2*10^5. - Tyler NeSmith, May 04 2021

			5^2 + 4 = 29 is prime, so 2 is a term.

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1

Cf. A059613.

Select[Range[0, 2000], PrimeQ[5^# + 4] &] (* Vincenzo Librandi, Oct 01 2012 *)

is(n)=ispseudoprime(5^n+4) \\ Charles R Greathouse IV, Feb 17 2017

Added the first term 0 by Vincenzo Librandi, Oct 01 2012
a(9) - a(10) from Bruno Berselli, Aug 12 2013
a(11)-a(12) from Robert Price, Feb 03 2014
a(13) from Tyler NeSmith, May 04 2021

A191469 Numbers n such that 7^n - 6 is prime.

Original entry on oeis.org

2, 3, 6, 9, 21, 25, 33, 49, 54, 133, 245, 255, 318, 1023, 1486, 3334, 6821, 8555, 11605, 42502, 44409, 90291, 92511, 140303

Offset: 1

Vincenzo Librandi, Jun 06 2011

a(14)=1023 and a(15)=1486 correspond to BPSW strong probable primes (passing PARI's ispseudoprime()). - Joerg Arndt, Jun 06 2011

a(25) > 2*10^5. - Robert Price, Nov 14 2014

Cf. A014224, A059266, A059613, A059614, A095714, A177093.

Cf. A236371, A217131, A191469, A090669, A096305, A217130, A217132, A152213.

Magma
```
[n: n in [1..1000]| IsPrime(7^n-6)]
```

A191469:=n->`if`(isprime(7^n-6),n,NULL): seq(A191469(n), n=1..10^3); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[1,5000],PrimeQ[7^#-6]&] (* Vincenzo Librandi, Aug 05 2012 *)

for(n=1, 10^6, if(isprime(7^n-6), print1(n, ", ")))

a(17)-a(23) from Robert Price, Jan 24 2014

a(24) from Robert Price, Nov 14 2014

A220290 4-hyperperfect numbers: n = 4*(sigma(n)-n-1) + 1.

Original entry on oeis.org

1950625, 1220640625, 186264514898681640625

Offset: 1

Arkadiusz Wesolowski, Dec 11 2012

For all k in A059613, (5^k-4)*5^(k-1) is a term. In particular, k=15 gives a term 186264514898681640625.

Judson S. McCranie, A Study of Hyperperfect Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.3.
Eric Weisstein's World of Mathematics, Hyperperfect Number

Cf. A000396, A007593, A028499-A028502, A034916.

Select[Range[1, 2*10^6, 4], # + 3 == 4*(DivisorSigma[1, #] - #) &]

a(3) from Max Alekseyev, Jun 01 2025

A164785 a(n) = 5^n - 4.

Original entry on oeis.org

1, 21, 121, 621, 3121, 15621, 78121, 390621, 1953121, 9765621, 48828121, 244140621, 1220703121, 6103515621, 30517578121, 152587890621, 762939453121, 3814697265621, 19073486328121, 95367431640621, 476837158203121

Offset: 1

Daniel Minoli (daniel.minoli(AT)ses.com), Aug 26 2009

Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below: - Sequence m(n,t) = (n^t) - (n-1) for t=2 to infinity is called a Mersenne Sequence Rooted on n - If n is prime, this sequence is called a Legitimate Mersenne Sequence - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN) - If j belonging to the sequence m (n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP). Note: m (n,t) = n* m (n,t-1) + n^2 - 2*n+1. This sequence related to sequences: A014232 and A014224; A135535 and A059266. These numbers play a role in the context of hyperperfect numbers. For additional references, beyond key ones listed below, see A164783.

Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (p.114-134)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Daniel Minoli and W. Nakamine, Mersenne Numbers Rooted On 3 For Number Theoretic Transforms, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
Daniel Minoli and Robert Bear, Hyperperfect Numbers, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A059613.

[5^n-4: n in [1..30]]; // Vincenzo Librandi, Feb 06 2013

5^Range[50]-4 (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)
LinearRecurrence[{6,-5},{1,21},30] (* or *) NestList[5 # + 16 &, 1, 30] (* Harvey P. Dale, Jun 07 2012 *)
CoefficientList[Series[(1 + 15 x)/(1 - 6 x + 5 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Feb 06 2013 *)

a(n) = 5*a(n-1) + 16 with n > 1, a(1)=1. - Vincenzo Librandi, Nov 30 2010
a(n) = 6*a(n-1) - 5*a(n-2); a(1)=1, a(2)=21. - Harvey P. Dale, Jun 07 2012
G.f.: x*(1 + 15*x)/(1 - 6*x + 5*x^2). - Vincenzo Librandi, Feb 06 2013
E.g.f.: 3 + (exp(4*x) - 4)*exp(x). - Ilya Gutkovskiy, Jun 11 2016

More terms a(9)-a(21) from Vincenzo Librandi, Oct 29 2009

A181285 Primes of the form 5^k - 4.

Original entry on oeis.org

3121, 78121, 30517578121, 710542735760100185871124267578121, 413590306276513837435704346034981426782906055450439453121

Offset: 1

Jonathan D. B. Hodgson, Oct 12 2010

nonn

			3121 = 5^5 - 4 is prime and therefore is in the sequence.

Cf. A059613, A135535, A228028.

n:=1000: S:={}: for i from 1 to n do if type(5^i-4,prime)=true then S:=S union {5^i-4} end if od; S;

Select[5^Range[90]-4,PrimeQ] (* Harvey P. Dale, Aug 23 2013 *)

A217134 Numbers n such that 5^n - 8 is prime.

Original entry on oeis.org

2, 4, 10, 14, 88, 112, 140, 764, 3040, 11096, 24934, 25616, 54584, 93400

Offset: 1

Vincenzo Librandi, Oct 01 2012

nonn

a(15) > 10^5. - Robert Price, Feb 03 2014

Cf. A059613, A087885, A089142, A109080, A124621, A165701.

Select[Range[2, 5000], PrimeQ[5^# - 8] &]

for(n=2, 5*10^3, if(isprime(5^n-8), print1(n", ")))

a(10)-a(14) from Robert Price, Feb 03 2014

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1

Offset: 2

Eric Chen, Jun 04 2018

nonn

a(prime(j)) + 1 = A087139(j).
a(123) > 10^5, a(342) > 10^5, see the Barnes link for the Sierpinski base-123 and base-342 problems.
a(251) > 73000, see A087139.

Eric Chen, Table of n, a(n) for n = 2..122
Gary Barnes, Sierpinski conjectures and proofs
Eric Chen, Table n, a(n) for n = 2..360 status

For the numbers k such that these forms are prime:
a1(b): numbers k such that (b-1)*b^k-1 is prime
a2(b): numbers k such that (b-1)*b^k+1 is prime
a3(b): numbers k such that (b+1)*b^k-1 is prime
a4(b): numbers k such that (b+1)*b^k+1 is prime (no such k exists when b == 1 (mod 3))
a5(b): numbers k such that b^k-(b-1) is prime
a6(b): numbers k such that b^k+(b-1) is prime
a7(b): numbers k such that b^k-(b+1) is prime
a8(b): numbers k such that b^k+(b+1) is prime (no such k exists when b == 1 (mod 3)).
Using "-------" if there is currently no OEIS sequence and "xxxxxxx" if no such k exists (this occurs only for a4(b) and a8(b) for b == 1 (mod 3)):
.
   b   a1(b)   a2(b)   a3(b)   a4(b)   a5(b)   a6(b)   a7(b)   a8(b)
--------------------------------------------------------------------
   2 A000043 ------- A002235 A002253 A000043 ------- A050414 A057732
   3 A003307 A003306 A005540 A005537 A014224 A051783 A058959 A058958
   4 A272057 ------- ------- xxxxxxx A059266 A089437 A217348 xxxxxxx
   5 A046865 A204322 A257790 A143279 A059613 A124621 A165701 A089142
   6 A079906 A247260 ------- ------- A059614 A145106 A217352 A217351
   7 A046866 A245241 ------- xxxxxxx A191469 A217130 A217131 xxxxxxx
   8 A268061 A269544 ------- ------- A217380 A217381 A217383 A217382
   9 A268356 A056799 ------- ------- A177093 A217385 A217493 A217492
  10 A056725 A056797 A111391 xxxxxxx A095714 A088275 A092767 xxxxxxx
  11 A046867 A057462 ------- ------- ------- ------- ------- -------
  12 A079907 A251259 ------- ------- ------- A137654 ------- -------
  13 A297348 ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
  14 A273523 ------- ------- ------- ------- ------- ------- -------
  15 ------- ------- ------- ------- ------- ------- ------- -------
  16 ------- ------- ------- xxxxxxx ------- ------- ------- xxxxxxx
Cf. (smallest k such that these forms are prime) A122396 (a1(b)+1 for prime b), A087139 (a2(b)+1 for prime b), A113516 (a5(b)), A076845 (a6(b)), A178250 (a7(b)).

a(n)=for(k=1,2^16,if(ispseudoprime((n-1)*n^k+1),return(k)))

A174267 Primes p such that 5^p - 4 is also prime.

Original entry on oeis.org

5, 7, 47

Offset: 1

Vincenzo Librandi, Mar 14 2010

If it exists, a(4) > 48091. - J.W.L. (Jan) Eerland, Nov 13 2022

If it exists, a(4) > 2*10^5 (cf. comment by Tyler NeSmith in A059613). - Michael S. Branicky, Aug 17 2024

			For p=5, 5^5-4=3121; p=7, 5^7-4=78121; p=47, 5^47-4= 710542735760100185871124267578121

Cf. A000040, A059613.

[p: p in PrimesUpTo(3500) | IsPrime(5^p-4)];

is(n)=isprime(n) && isprime(5^n-4) \\ Charles R Greathouse IV, Sep 06 2016

A000040 INTERSECT A059613. - R. J. Mathar, Apr 15 2010

A378868 Numbers k such that 5^k - 22 is prime.

Original entry on oeis.org

2, 3, 31, 79, 491, 3019, 3623, 4175, 9957, 21963, 71637, 80551, 80831

Offset: 1

Robert Price, Dec 09 2024

a(14) > 10^5. - Michael S. Branicky, Dec 24 2024

			3 is a term because 5^3 - 22 = 103 is prime.

Cf. A059613, A109080, A165701, A217134.

Magma
```
[k: k in [0..1000] |IsPrime (5^k-22)];
```
Mathematica
```
Select[Range[0,5000],PrimeQ[5^#-22]&]
```

a(8)-a(10) from Michael S. Branicky, Dec 17 2024

a(11)-a(13) from Michael S. Branicky, Dec 22 2024

cp's OEIS Frontend

A124621 Numbers k such that 5^k + 4 is prime.

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A191469 Numbers n such that 7^n - 6 is prime.

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A220290 4-hyperperfect numbers: n = 4*(sigma(n)-n-1) + 1.

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A164785 a(n) = 5^n - 4.

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A181285 Primes of the form 5^k - 4.

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Maple

Mathematica

A217134 Numbers n such that 5^n - 8 is prime.

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Mathematica

PARI

Extensions

A305531 Smallest k >= 1 such that (n-1)*n^k + 1 is prime.

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Comments

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Crossrefs

Programs

PARI

A174267 Primes p such that 5^p - 4 is also prime.

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Author

Keywords