A007658 -id:A007658 - cp's OEIS frontend

A128067 Numbers k such that (3^k + 7^k)/10 is prime.

3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.

a(10) > 10^5. - Robert Price, Oct 03 2012

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128068, A128069, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=7; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

is(n)=isprime((3^n+7^n)/10) \\ Charles R Greathouse IV, Feb 17 2017

More terms from Ryan Propper, Apr 02 2007

a(7)-a(9) from Robert Price, Oct 03 2012

A128069 Numbers k such that (3^k + 10^k)/13 is prime.

Original entry on oeis.org

3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.
Next term is greater than 6700. - Stefan Steinerberger, May 11 2007
a(11) > 10^5. - Robert Price, Jan 15 2013

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128067, A128068, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=10; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

is(n)=isprime((3^n+10^n)/13) \\ Charles R Greathouse IV, Feb 17 2017

a(7) from Alexander Adamchuk, Feb 14 2007

a(8)-a(10) from Robert Price, Jan 15 2013

A128070 Numbers k such that (3^k + 11^k)/14 is prime.

Original entry on oeis.org

3, 103, 271, 523, 23087, 69833

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.

a(7) > 10^5. - Robert Price, Mar 04 2013

Cf. A007658 (numbers k such that (3^k + 1)/4 is prime).
Cf. A057469 (numbers k such that (3^k + 2^k)/5 is prime).
Cf. A122853 (numbers k such that (3^k + 5^k)/8 is prime).
Cf. A128066, A128067, A128068, A128069, A128071, A128072, A128073, A128074, A128075.
Cf. A059801 (numbers k such that 4^k - 3^k is prime).
Cf. A121877 (numbers k such that (5^k - 3^k)/2 is prime).
Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=11; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

is(n)=isprime((3^n+11^n)/14) \\ Charles R Greathouse IV, Feb 17 2017

a(5)-a(6) from Robert Price, Mar 04 2013

A128068 Numbers k such that (3^k + 8^k)/11 is prime.

Original entry on oeis.org

5, 163, 191, 229, 271, 733, 21059, 25237

Offset: 1

Alexander Adamchuk, Feb 14 2007

All terms are primes.

a(9) > 10^5. - Robert Price, Mar 06 2013

Cf. A007658 = numbers n such that (3^n + 1)/4 is prime. Cf. A057469 = numbers n such that (3^n + 2^n)/5 is prime. Cf. A122853 = numbers n such that (3^n + 5^n)/8 is prime. Cf. A128066, A128067, A128069, A128070, A128071, A128072, A128073, A128074, A128075. Cf. A059801 = numbers n such that 4^n - 3^n is prime. Cf. A121877 = numbers n such that (5^n - 3^n)/2 is a prime. Cf. A128024, A128025, A128026, A128027, A128028, A128029, A128030, A128031, A128032.

k=8; Do[ p=Prime[n]; f=(3^p+k^p)/(k+3); If[ PrimeQ[f], Print[p]], {n,1,100} ]

is(n)=isprime((3^n+8^n)/11) \\ Charles R Greathouse IV, Feb 17 2017

a(6) from Alexander Adamchuk, Feb 14 2007

a(7)-a(8) from Robert Price, Mar 06 2013

A229524 Numbers k such that (38^k + 1)/39 is prime.

Original entry on oeis.org

5, 167, 1063, 1597, 2749, 3373, 13691, 83891, 131591

Offset: 1

Robert Price, Sep 25 2013

All terms are primes. a(9) > 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. 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.

Do[ p=Prime[n]; If[ PrimeQ[ (38^p + 1)/39 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((38^n+1)/39) \\ Charles R Greathouse IV, Feb 17 2017

a(9)=131591 corresponds to a probable prime discovered by Paul Bourdelais, Jul 03 2018

A229663 Numbers n such that (40^n + 1)/41 is prime.

Original entry on oeis.org

53, 67, 1217, 5867, 6143, 11681, 29959

Offset: 1

Robert Price, Sep 27 2013

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.

Do[ p=Prime[n]; If[ PrimeQ[ (40^p + 1)/41 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((40^n+1)/41) \\ Charles R Greathouse IV, Feb 17 2017

A230036 Numbers n such that (39^n + 1)/40 is prime.

Original entry on oeis.org

3, 13, 149, 15377

Offset: 1

Robert Price, Oct 05 2013

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.

Do[ p=Prime[n]; If[ PrimeQ[ (39^p + 1)/40 ], Print[p] ], {n, 1, 9592} ]

is(n)=ispseudoprime((39^n+1)/40) \\ Charles R Greathouse IV, Feb 17 2017

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

Original entry on oeis.org

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

A111010 Primes of the form (3^k - (-1)^k)/4.

Original entry on oeis.org

2, 7, 61, 547, 398581, 23535794707, 82064241848634269407

Offset: 1

Cino Hilliard, Oct 02 2005

nonn

The next term is too large to include.
Is there an infinity of primes in this sequence?
All a(n), except a(1) = 2, are primes of the form (3^k + 1)/4. Corresponding numbers k such that (3^k + 1)/4 is prime are listed in A007658(n) = {3, 5, 7, 13, 23, 43, 281, 359, 487, 577, ...}. All such numbers k are primes. a(1) = 2 is the only prime of the form (3^k - 1)/4. - Alexander Adamchuk, Nov 19 2006

John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, p. 16.

Amiram Eldar, Table of n, a(n) for n = 1..14 (terms 1..11 from Alexander Adamchuk, Nov 19 2006)

Cf. A007658, A015518.

Do[f=(3^n - (-1)^n)/4; If[PrimeQ[f],Print[{n,f}]],{n,1,577}] (* Alexander Adamchuk, Nov 19 2006 *)

primenum(n,k,typ) = /* k=mult,typ=1 num,2 denom. ouyput prime num or denom. */ { local(a,b,x,tmp,v); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,v=a,v=b); if(isprime(v),print1(v","); ) ); print(); print(a/b+.); }

Given a(0)=1, b(0)=1, then for i=1, 2, ..., a(i)/b(i) = (a(i-1) + 2*b(i-1)) /(a(i-1) + b(i-1)).

a(n) = A015518(A007658(n-1)) for n >= 2. - Amiram Eldar, Jul 04 2024

Edited by Alexander Adamchuk, Nov 19 2006

cp's OEIS Frontend

A128067 Numbers k such that (3^k + 7^k)/10 is prime.

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A128069 Numbers k such that (3^k + 10^k)/13 is prime.

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A128070 Numbers k such that (3^k + 11^k)/14 is prime.

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A128068 Numbers k such that (3^k + 8^k)/11 is prime.

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A229524 Numbers k such that (38^k + 1)/39 is prime.

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A229663 Numbers n such that (40^n + 1)/41 is prime.

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A230036 Numbers n such that (39^n + 1)/40 is prime.

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

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