A204940 -id:A204940 - cp's OEIS frontend

A375161 Numbers k such that (23^k - 2^k)/21 is prime.

Original entry on oeis.org

5, 11, 197, 4159

Offset: 1

Robert Price, Aug 04 2024

The definition implies that k must be a prime.

a(5) > 10^5.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
P. Bourdelais, A Generalized Repunit Conjecture
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542.

Select[Prime[Range[100]], PrimeQ[(23^# - 2^#)/21] &]

A375236 Numbers k such that (21^k - 2^k)/19 is prime.

Original entry on oeis.org

2, 3, 353, 751, 9587

Offset: 1

Robert Price, Aug 06 2024

The definition implies that k must be a prime.

a(6) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161.

Select[Prime[Range[100]], PrimeQ[(21^# - 2^#)/19] &]

A377031 Numbers k such that (27^k - 2^k)/25 is prime.

Original entry on oeis.org

2, 3, 269, 401, 631, 701, 1321, 2707, 5471, 6581

Offset: 1

Robert Price, Oct 13 2024

The definition implies that k must be a prime.

a(11) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236.

Select[Prime[Range[1000]], PrimeQ[(27^# - 2^#)/25] &]

A377856 Numbers k such that (21^k + 2^k)/23 is prime.

Original entry on oeis.org

11, 17, 47, 2663

Offset: 1

Robert Price, Nov 09 2024

The definition implies that k must be a prime.

a(5) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A057187, A057188, A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A228922, A229542, A375161, A375236, A377031.

Select[Prime[Range[10000]], PrimeQ[(21^# + 2^#)/23] &]

A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

Original entry on oeis.org

3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3

Offset: 2

Alexander Adamchuk, Feb 20 2007

nonn

a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2.
All nonzero terms are odd primes.
a(n) = 0 for n = {4,9,16,25,32,36,49,64,81,100,121,125,144,...}, which are the perfect powers with exceptions of the form n^(p^m) where p>2 and (n^(p^(m+1))-1)/(n^(p^m)-1) are prime and m>=1 (in which case a(n^(p^m))=p). - Max Alekseyev, Jan 24 2009
a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
a(152) > 20000. - Eric Chen, Jun 01 2015
a(n) is the least number k such that (n^k - 1)/(n-1) is a Brazilian prime, or 0 if no such Brazilian prime exists. - Bernard Schott, Apr 23 2017
These corresponding Brazilian primes are in A285642. - Bernard Schott, Aug 10 2017
a(152) = 270217, see the top PRP link. - Eric Chen, Jun 04 2018
a(184) = 16703, a(200) = 17807, a(210) = 19819, a(306) = 26407, a(311) = 36497, a(326) = 26713, a(331) = 25033; a(185) > 66337, a(269) > 63659, a(281) > 63421, and there are 48 unknown a(n) for n <= 1024. - Eric Chen, Jun 04 2018
Six more terms found: a(522)=20183, a(570)=12907, a(684)=22573, a(731)=15427, a(820)=12043, a(996)=14629. - Michael Stocker, Apr 09 2020

			a(7) = 5 because (7^5 - 1)/6 = 2801 = 11111_7 is prime and (7^k - 1)/6 = 1, 8, 57, 400 for k = 1, 2, 3, 4. - _Bernard Schott_, Apr 23 2017

Max Alekseyev and Eric Chen, Table of n, a(n) for n = 2..184 (terms 2..151 from Max Alekseyev)
Eric Chen, Table of n, a(n) for n = 2..1024 status (updated by Jinyuan Wang)
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Richard Fischer, Generalized repunit primes of the form (B^N-1)/(B-1)
Top PRPs, Search by (152^n-1)/(152-1)
Top PRPs, Search by (b^n-1)/a
Eric Weisstein's World of Mathematics, Repunit

Cf. A084738, A065854, A084740, A084741, A065507, A084742, A066180, A084732, A285642, A085104.
Cf. A002384, A049409, A100330, A162862, A217070-A217089. (numbers b such that (b^p-1)/(b-1) is prime for prime p = 3 to 97)
Cf. A000043, A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A133857, A006035, A127995, A127996, A127997, A204940, A127998, A127999, A128000, A181979, A098438, A128002, A209120, A185073, A128003, A128004, A181987, A128005, A239637, A240765, A294722, A242797, A243279, A267375, A245237, A245442, A173767. (numbers n such that (b^n-1)/(b-1) is prime for b = 2 to 53)
A126589 gives locations of zeros.

Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[(m^k - 1)/(m - 1)], k++]; k, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* Michael De Vlieger, Apr 24 2017 *)

a052409(n) = my(k=ispower(n)); if(k, k, n>1)
a052410(n) = if (ispower(n, , &r), r, n)
is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
a(n) = if(is(n), 0, forprime(p=3, 2^16, if(ispseudoprime((n^p-1)/(n-1)), return(p)))) \\ Eric Chen, Jun 01 2015, corrected by Eric Chen, Jun 04 2018, after Charles R Greathouse IV in A052409 and Michel Marcus in A052410

a(18) = 25667 found by Henri Lifchitz, Sep 26 2007

A376329 Numbers k such that (45^k - 2^k)/43 is prime.

Original entry on oeis.org

2, 7, 89, 167, 8101, 96517

Offset: 1

Robert Price, Nov 19 2024

The definition implies that k must be a prime.

a(7) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236, A377031.

Select[Prime[Range[10000]], PrimeQ[(45^# - 2^#)/43] &]

A376470 Numbers k such that (29^k - 2^k)/27 is prime.

Original entry on oeis.org

2, 7, 139, 983, 3257, 10181, 26387, 36187, 42557

Offset: 1

Robert Price, Sep 24 2024

The definition implies that k must be a prime.

a(10) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236.

Select[Prime[Range[1000]], PrimeQ[(29^# - 2^#)/27] &]

A377180 Numbers k such that (43^k - 2^k)/41 is prime.

Original entry on oeis.org

167, 797, 1009, 54941

Offset: 1

Robert Price, Oct 18 2024

The definition implies that k must be a prime.

a(5) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236, A377031.

Select[Prime[Range[10000]], PrimeQ[(43^# - 2^#)/41] &]

A377699 Numbers k such that (35^k - 2^k)/33 is prime.

Original entry on oeis.org

2, 17, 53, 211, 4013, 55207

Offset: 1

Robert Price, Nov 05 2024

The definition implies that k must be a prime.

a(7) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236, A377031.

Select[Prime[Range[10000]], PrimeQ[(35^# - 2^#)/33] &]

A377718 Numbers k such that (41^k - 2^k)/39 is prime.

Original entry on oeis.org

2, 41, 97, 131, 2411, 7321

Offset: 1

Robert Price, Nov 04 2024

The definition implies that k must be a prime.

a(7) > 10^5.

P. Bourdelais, A Generalized Repunit Conjecture.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit.

Cf. A062587, A062589, A127996, A127997, A128344, A204940, A217320, A225807, A229542, A375161, A375236, A377031.

Select[Prime[Range[10000]], PrimeQ[(41^# - 2^#)/39] &]

cp's OEIS Frontend

A375161 Numbers k such that (23^k - 2^k)/21 is prime.

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A375236 Numbers k such that (21^k - 2^k)/19 is prime.

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A377031 Numbers k such that (27^k - 2^k)/25 is prime.

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A377856 Numbers k such that (21^k + 2^k)/23 is prime.

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A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

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A376329 Numbers k such that (45^k - 2^k)/43 is prime.

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A376470 Numbers k such that (29^k - 2^k)/27 is prime.

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A377180 Numbers k such that (43^k - 2^k)/41 is prime.

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A377699 Numbers k such that (35^k - 2^k)/33 is prime.

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