A065507 -id:A065507 - cp's OEIS frontend

A231604 Numbers n such that (42^n + 1)/43 is prime.

3, 709, 1637, 17911, 127609, 172663

Offset: 1

Robert Price, Nov 11 2013

The first 5 terms are primes.

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. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663.

Do[ p=Prime[n]; If[ PrimeQ[ (42^p + 1)/43 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((42^n+1)/43) \\ Charles R Greathouse IV, Feb 20 2017

a(5)=127609 corresponds to a probable prime discovered by Paul Bourdelais, Jul 02 2018

a(6)=172663 corresponds to a probable prime discovered by Paul Bourdelais, Jul 29 2019

A231865 Numbers n such that (43^n + 1)/44 is prime.

Original entry on oeis.org

5, 7, 19, 251, 277, 383, 503, 3019, 4517, 9967, 29573

Offset: 1

Robert Price, Nov 14 2013

All terms are primes.

a(11) > 10^5.

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. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604.

Do[ p=Prime[n]; If[ PrimeQ[ (43^p + 1)/44 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((43^n+1)/44) \\ Charles R Greathouse IV, Feb 20 2017

A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 43, 61, 157, 521, 547, 683, 2731, 4423, 6163, 13421, 19183, 22651, 26407, 37057, 43691, 113233, 121453, 143263, 174763, 208393, 292141, 375157, 398581, 412807, 527803, 590593, 843643, 981091, 1041421, 1193557, 1246573

Offset: 1

T. D. Noe, Jan 25 2007

These primes are important in studying k-imperfect numbers (A127724), see Iannucci-link. Except for the cases p^e = 3 and 8, which yield primes 2 and 5, e is an even number such that e+1 is prime. In fact, except for those two cases, all the primes are of the form (1+p^q)/(1+p), where q is an odd prime; that is, repunit primes with negative prime base.

			From _David A. Corneth_, Oct 28 2017: (Start)
For (p, e) = (3, 1) we have the prime 3^1 - 3^0 = 2.
For (p, e) = (2, 3) we have the prime 2^3 - 2^2 + 2^1 - 2^0 = 5.
The examples above are the cases mentioned in the comments not of the form (1+p^q)/(1+p). A prime of that form is below;
For (p, e) = (2, 4) we have the prime 2^4 - 2^3 + 2^2 - 2^1 + 2^0 = 11 = (1+p^(e+1)) / (1+p) = 33/3. (End)

David A. Corneth, Table of n, a(n) for n = 1..33914 (terms < 10^14) (the first 4799 terms < 10^12 from T. D. Noe)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), Article 00.2.7.
Douglas E. Iannucci, On a variation of perfect numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory, 6 (2006), #A41.

Cf. A023195, A065507, A084741, A206369.

upto(n) = {my(res = List([2,5])); forprime(p = 2, sqrtnint(n, 2), forprime(q = 3, logint(n * (1+p), p), r = (1+p^q)/(1+p); if(isprime(r), listput(res, r)))); listsort(res, 1); res} \\ David A. Corneth, Oct 28 2017

A235683 Numbers n such that (46^n + 1)/47 is prime.

Original entry on oeis.org

7, 23, 59, 71, 107, 223, 331, 2207, 6841, 94841

Offset: 1

Robert Price, Jan 13 2014

All terms up to a(10) are primes.

a(11) > 10^5.

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. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865.

Do[ p=Prime[n]; If[ PrimeQ[ (46^p + 1)/47 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((46^n+1)/47) \\ Charles R Greathouse IV, May 22 2017

A237052 Numbers n such that (49^n + 1)/50 is prime.

Original entry on oeis.org

7, 19, 37, 83, 1481, 12527, 20149

Offset: 1

Robert Price, Feb 02 2014

All terms are primes.

a(8) > 10^5.

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. A000978 = numbers n such that (2^n + 1)/3 is prime.

Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683, A236167, A236530.

Do[ p=Prime[n]; If[ PrimeQ[ (49^p + 1)/50 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((49^n+1)/50) \\ Charles R Greathouse IV, Jun 13 2017

Typo in description corrected by Ray Chandler, Feb 20 2017

A126589 Numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2; or A128164(n) = 0.

Original entry on oeis.org

4, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 125, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1728, 1764, 1849, 1936, 2025

Offset: 1

Alexander Adamchuk, Mar 13 2007

nonn

Appears to be the union of the perfect squares k^2 (for k>1) and the prime powers p^k (for k>1) with some exceptions, such as 2^3, 3^3, 2^7, etc.

The perfect powers except those of the form n^(p^m) where p and (n^(p^(m+1))-1)/(n^(p^m)-1) are primes, p>2 and m>=1. - Max Alekseyev, Mar 09 2009

			A128164 begins with offset 2: {3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, ...}. Thus a(1) = 4, a(2) = 9, a(3) = 16.

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Eric Weisstein's World of Mathematics, Repunit.

Cf. A128164, A084738, A065854, A084740, A084741, A065507, A084742.

Extended by Max Alekseyev, Mar 09 2009

A236167 Numbers k such that (47^k + 1)/48 is prime.

Original entry on oeis.org

5, 19, 23, 79, 1783, 7681

Offset: 1

Robert Price, Jan 19 2014

a(7) > 10^5.

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. A000978 = numbers k such that (2^k + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683.

Do[ p=Prime[n]; If[ PrimeQ[ (47^p + 1)/48 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((47^n+1)/48) \\ Charles R Greathouse IV, Jun 06 2017

from sympy import isprime
def afind(startat=0, limit=10**9):
  pow47 = 47**startat
  for k in range(startat, limit+1):
    q, r = divmod(pow47+1, 48)
    if r == 0 and isprime(q): print(k, end=", ")
    pow47 *= 47
afind(limit=300) # Michael S. Branicky, May 19 2021

A185230 Numbers n such that (33^n + 1)/34 is prime.

Original entry on oeis.org

5, 67, 157, 12211, 313553

Offset: 1

Robert Price, Aug 29 2013

All terms are 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.
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. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382. Cf. A084741, A084742, A065507, A126659, A126856.

Do[ p=Prime[n]; If[ PrimeQ[ (33^p + 1)/34 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((33^n+1)/34) \\ Charles R Greathouse IV, Jun 13 2017

a(5) from Paul Bourdelais, Feb 26 2021

A236530 Numbers n such that (48^n + 1)/49 is prime.

Original entry on oeis.org

5, 17, 131, 84589

Offset: 1

Robert Price, Jan 27 2014

All terms are primes.

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.
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. A000978 = numbers n such that (2^n + 1)/3 is prime. Cf. A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A084741, A084742, A065507, A126659, A126856, A185240, A229145, A229524, A230036, A229663, A231604, A231865, A235683, A236167.

Do[ p=Prime[n]; If[ PrimeQ[ (48^p + 1)/49 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((48^n+1)/49) \\ Charles R Greathouse IV, Jun 13 2017

Incorrect first term deleted by Robert Price, Feb 21 2014

cp's OEIS Frontend

A231604 Numbers n such that (42^n + 1)/43 is prime.

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A231865 Numbers n such that (43^n + 1)/44 is prime.

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A127727 Primes of the form p^e - p^(e-1) + p^(e-2) - ... + (-1)^e, where p is prime.

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A235683 Numbers n such that (46^n + 1)/47 is prime.

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A237052 Numbers n such that (49^n + 1)/50 is prime.

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A126589 Numbers n>1 such that prime of the form (n^k-1)/(n-1) does not exist for k>2; or A128164(n) = 0.

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A236167 Numbers k such that (47^k + 1)/48 is prime.

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A185230 Numbers n such that (33^n + 1)/34 is prime.

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