A001562 -id:A001562 - 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

A054416 Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.

Original entry on oeis.org

2, 3, 9, 15, 26, 33, 146, 320, 1068, 1505, 134103, 800393

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000

Numbers k such that 10*(10^(2k)-1)/11 + 1 is prime.

			The first 3 numbers are 9091, 909091, 909090909090909091.

J. A. H. Hunter and J. S. Madachy, Mathematical Diversions, New York: Dover Publications, Inc., 1974, pp. 4-5. Originally published by Van Nostrand, 1963.

David Broadhurst, Proof that 1505 term is prime [BROKEN LINK]
David Broadhurst, Unique record, digest of 3 messages in primenumbers Yahoo group, Apr 8-9, 2001. [Cached copy]

Cf. A001562.

Do[ If[ PrimeQ[ 10*(10^(2n) - 1)/11 + 1], Print[ n ] ], {n, 0, 1505} ]
Position[Table[FromDigits[PadLeft[{9,1},2n,{9,0}]],{n,1510}], ?PrimeQ]// Flatten (* _Harvey P. Dale, Nov 02 2017 *)

from sympy import isprime, prime
def afind(limit, startk=1):
    s = "90"*(startk-1)
    for k in range(startk, limit+1):
        if isprime(int(s+"91")):
            print(k, end=", ")
        s += "90"
afind(400) # Michael S. Branicky, Jan 13 2022

a(n) = (A001562(n)-1)/2.

More terms from Michael Kleber and Harvey Dubner (harvey(AT)dubner.com), May 22 2000
Ignacio Larrosa Cañestro reports that the 1068 term has now been established to be a prime using Titanix 1.01, Oct 23 2000
a(11)-a(12) from Michael S. Branicky, Jan 13 2022 using A001562

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

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

A309533 Numbers k such that (144^k + 1)/145 is prime.

Original entry on oeis.org

23, 41, 317, 3371, 45259, 119671

Offset: 1

Paul Bourdelais, Aug 06 2019

The corresponding primes are terms of A059055. - Bernard Schott, Aug 09 2019

Cf. A000978, A007658, A057171, A057172, A057173, A057175, A001562, A057177, A057178, A057179, A057180, A057181, A057182, A057183, A057184, A057185, A057186, A057187, A057188, A057189, A057190, A057191, A071380, A071381, A071382, A126856, A185240.

Do[p=Prime[n]; If[PrimeQ[(144^p + 1)/145], Print[p]], {n, 1, 1000000}]

PARI
```
is(n)=ispseudoprime((144^n+1)/145)
```

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

A215804 Odd numbers k such that 10^k + 1 can be written in the form a^2 + 2*b^2.

Original entry on oeis.org

1, 5, 7, 13, 19, 25, 29, 31, 35, 43, 53, 61, 65, 67, 71, 95, 125, 145, 155, 175, 179, 215, 239, 263, 265, 269, 293, 305

Offset: 1

V. Raman, Aug 23 2012

nonn

These 10^k + 1 numbers have no prime factors of the form 5 or 7 (mod 8) to an odd power.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200

Cf. A000533, A001562, A062397, A215805.

for(i=2, 100, a=factorint(10^i+1)~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i" -\t"a[1, ])))

12 more terms from V. Raman, Aug 28 2012

A215805 Prime numbers p such that 10^p + 1 can be written in the form a^2 + 2*b^2.

Original entry on oeis.org

5, 7, 13, 19, 29, 31, 43, 53, 61, 67, 71, 179, 239, 263, 269, 293

Offset: 1

V. Raman, Aug 23 2012

nonn

These numbers have no prime factors of the form 5 or 7 (mod 8) to an odd power.

Samuel S. Wagstaff, Jr., The Cunningham Project, Factorizations of 2^n-1, for odd n's < 1200.

Cf. A000533, A001562, A062397, A215804.

forprime(i=2, 100, a=factorint(10^i+1)~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i" -\t"a[1, ])))

5 more terms from V. Raman, Aug 29 2012

cp's OEIS Frontend

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

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A054416 Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.

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A231865 Numbers n such that (43^n + 1)/44 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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A309533 Numbers k such that (144^k + 1)/145 is prime.

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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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