A000978 -id:A000978 - cp's OEIS frontend

A126614 a(n) = (2^prime(n) + 1)/3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 178956971, 715827883, 45812984491, 733007751851, 2932031007403, 46912496118443, 3002399751580331, 192153584101141163, 768614336404564651, 49191317529892137643, 787061080478274202283, 3148244321913096809131

Offset: 2

Artur Jasinski, Feb 09 2007

nonn

If p - 1 is squarefree, the multiplicative order of 2 modulo a(n) is 2p. - Vladimir Shevelev, Jul 15 2008

The prime numbers in this sequence are the Wagstaff primes (A000979). - Omar E. Pol, Nov 05 2013

			a(2) = (2^prime(2) + 1)/3 = (2^3 + 1)/3 = 9/3 = 3.
a(3) = (2^prime(3) + 1)/3 = (2^5 + 1)/3 = 33/3 = 11.
a(4) = (2^prime(4) + 1)/3 = (2^7 + 1)/3 = 129/3 = 43.

Vincenzo Librandi, Table of n, a(n) for n = 2..200

Cf. A000040, A000979, A000978, A001045, A124400.

[(2^NthPrime(n)+1)/3: n in [2..20]]; // Vincenzo Librandi, Mar 29 2012

Mathematica
```
Table[(2^Prime[n] + 1)/3, {n, 2, 20}]
```

a(n)=2^prime(n)\/3 \\ Charles R Greathouse IV, Mar 29 2012

vecextract(apply(p->2^p\/3,primes(100)),"2..") \\ Charles R Greathouse IV, Mar 29 2012

a(n) = A001045(A000040(n)). - Alois P. Heinz, Apr 14 2025

A107036 Indices of prime Jacobsthal numbers.

Original entry on oeis.org

3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399

Offset: 1

Paul Barry, May 09 2005

All terms are prime except a(1) = 4. All prime terms are listed in A000978. - Alexander Adamchuk, Oct 03 2006

Cf. A000978, A001045, A001605.

is(n)=ispseudoprime((2^n - (-1)^n) / 3) \\ Charles R Greathouse IV, Apr 28 2015

a(n) = A000978(n-1) for n >= 3. - Alexander Adamchuk, Oct 03 2006

More terms from Alexander Adamchuk, Oct 03 2006

a(41)-a(42) from Bill McEachen, Aug 28 2024

A126856 Numbers n such that (31^n + 1)/32 is prime.

Original entry on oeis.org

109, 461, 1061, 50777

Offset: 1

Alexander Adamchuk, Mar 23 2007

All terms are primes.

a(5) > 10^5. - Robert Price, Jul 12 2013

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.
Eric Weisstein's World of Mathematics, Repunit.
H. Lifchitz, Mersenne and Fermat primes field

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.

Do[ p=Prime[n]; If[ PrimeQ[ (31^p + 1)/32 ], Print[p] ], {n,1,1100} ]

is(n)=isprime((31^n+1)/32) \\ Charles R Greathouse IV, Feb 17 2017

a(4) from Robert Price, Jul 12 2013

A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 15, 21, 30, 39, 50, 63, 83, 95, 99, 156, 173, 350, 854, 1308, 1769, 2903, 5250, 5345, 5639, 6195, 7239, 21368, 41669, 47684, 58619, 63515, 69468, 70539, 133508, 134993, 187160, 493095

Offset: 1

Artur Jasinski, Feb 08 2007, Feb 09 2007

If this sequence is infinite then so is A124401.
Equals A127965(n)/2.
The sum has the simple closed form 1 + 2/3*(4^n-1). - Stefan Steinerberger, Nov 24 2007
Terms beyond a(30) correspond to probable primes, cf. A000978. - M. F. Hasler, Aug 29 2008

			a(1)=1 because 1 + 2 = 3 is prime;
a(2)=2 because 1 + 2 + 2^3 = 11 is prime;
a(3)=3 because 1 + 2 + 2^3 + 2^5 = 43 is prime;
a(4)=5 because 1 + 2 + 2^3 + 2^5 + 2^7 + 2^9 = 683 is prime;
...

Cf. A127962, A127963, A127964, A127965, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936, A127936, A124401, A010051, A007583.

import Data.List (findIndices)
a127936 n = a127936_list !! (n-1)
a127936_list = findIndices ((== 1) . a010051'') a007583_list
-- Reinhard Zumkeller, Mar 24 2013

a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]], AppendTo[a, x]], {x, 1, 1000}]; a
b = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c], AppendTo[b, c]], {x, 0, 1000}]; a = {}; Do[AppendTo[a, FromDigits[IntegerDigits[b[[x]], 2]]], {x, 1, Length[b]}]; d = {}; Do[AppendTo[d, (1/2)(DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1])], {x, 1, Length[a]}]; d
Position[Accumulate[2^(2*Range[1000]-1)],?(PrimeQ[#+1]&)]//Flatten (* The program generates the first 21 terms of the sequence. To generate more, increase the Range constant. *) (* _Harvey P. Dale, Mar 23 2022 *)

for(n=1,999, ispseudoprime(2^(2*n+1)\3+1) & print1(n",")) \\ M. F. Hasler, Aug 29 2008

from sympy import isprime
A127936 = [i for i in range(1,10**3) if isprime(int('01'*i+'1',2))]
# Chai Wah Wu, Sep 05 2014

a(n) = floor(A000978(n)/2) = ceiling(log(4)(A000979(n))); A000978(n) = 2 a(n) + 1; A000979(n) = (2*4^a(n)+1)/3. - M. F. Hasler, Aug 29 2008

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 11 2007
2 more terms from Stefan Steinerberger, Nov 24 2007
6 more terms from Dmitry Kamenetsky, Jul 12 2008
a(30)-a(40) calculated from A000978 by M. F. Hasler, Aug 29 2008

A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

Original entry on oeis.org

178956971, 45812984491, 733007751851, 46912496118443, 3002399751580331, 192153584101141163, 49191317529892137643, 787061080478274202283, 3148244321913096809131, 3223802185639011132549803

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

If p-1 is squarefree, the terms are overpseudoprimes (see A141232). - Vladimir Shevelev, Jul 15 2008

G. C. Greubel, Table of n, a(n) for n = 1..440

Cf. A000979, A000978, A124400, A126614, A127956, A127957.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 2, 30}]; a
Select[(2^Prime[Range[2,30]]+1)/3,CompositeQ] (* Harvey P. Dale, Feb 04 2015 *)

A127957 Numbers k such that (2^prime(k) + 1)/3 is composite.

Original entry on oeis.org

10, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1175

Cf. A000979, A000978, A124400, A126614, A127955, A127956.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, x]], {x, 2, 300}]; a
Select[Range[2,100],!PrimeQ[(2^Prime[#]+1)/3]&] (* Harvey P. Dale, Nov 24 2013 *)

isok(n) = (n!=1) && !isprime((2^prime(n)+1)/3); \\ Michel Marcus, Jul 07 2018

A127956 Prime numbers p such that (2^p+1)/3 is composite.

Original entry on oeis.org

29, 37, 41, 47, 53, 59, 67, 71, 73, 83, 89, 97, 103, 107, 109, 113, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 193, 197, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 317, 331, 337, 349, 353, 359, 367, 373

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

If p-1 is squarefree, 2a(n) is the multiplicative order of 2 modulo every divisor d>1 of (2^p+1)/3. - Vladimir Shevelev, Jul 15 2008

Cf. A000979, A000978, A124400, A126614, A127955, A127957.

a = {}; Do[c = (2^Prime[x] + 1)/3; If[PrimeQ[c] == False, AppendTo[a, Prime[x]]], {x, 2, 100}]; a
Select[Prime[Range[2,100]],CompositeQ[(2^#+1)/3]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

A229145 Numbers k such that (36^k + 1)/37 is prime.

Original entry on oeis.org

31, 191, 257, 367, 3061, 110503, 1145393

Offset: 1

Robert Price, Sep 15 2013

All such numbers k are prime.
Note that a(6) = 110503 corresponds to (36^110503 + 1)/37, which is only a probable prime with 171975 digits.
The primes corresponding to the terms of this sequence have 1 as their last digit and an even number as their next-to-last digit. - Iain Fox, Dec 08 2017

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.

Do[ p=Prime[n]; If[ PrimeQ[ (36^p + 1)/37 ], Print[p] ], {n, 1, 9592} ]

is(n)=isprime((36^n+1)/37) \\ Charles R Greathouse IV, Feb 17 2017

a(6) = 110503 (posted by Lelio R. Paula on primenumbers.net) from Paul Bourdelais, Dec 08 2017

a(7) from Paul Bourdelais, Nov 03 2023

A127958 Numbers x such that 1 + Sum_{k=1..n} 2^(2k-1) is not prime for n=1,2,...,x.

Original entry on oeis.org

4, 7, 10, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 84, 85

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

Numbers x such that 1 + Sum_{k=1..n} 2^(2k-1) is prime for n=1,2,...,x gives A127936.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000979, A000978, A124400, A126614, A127955, A127957, A127936.

a = {}; Do[If[PrimeQ[1 + Sum[2^(2n - 1), {n, 1, x}]] == False, AppendTo[a, x]], {x, 1, 1000}]; a

isok(x) = !isprime(1+sum(k=1, x, 2^(2*k-1))); \\ Michel Marcus, May 09 2018

A185240 Numbers k such that (35^k + 1)/36 is prime.

Original entry on oeis.org

11, 13, 79, 127, 503, 617, 709, 857, 1499, 3823, 135623, 280979

Offset: 1

Robert Price, Aug 29 2013

All terms are primes. a(11) > 10^5.

Paul 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. Cf. A084741, A084742, A065507, A126659, A126856.

Do[ p=Prime[n]; If[ PrimeQ[ (35^p + 1)/36 ], Print[p] ], {n, 1, 9592} ]

is(n)=isprime((35^n+1)/36) \\ Charles R Greathouse IV, Feb 17 2017

a(11)=135623 found as probable prime and added by Paul Bourdelais, Jul 05 2018

a(12) from Paul Bourdelais, Sep 13 2021

cp's OEIS Frontend

A126614 a(n) = (2^prime(n) + 1)/3.

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A107036 Indices of prime Jacobsthal numbers.

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A126856 Numbers n such that (31^n + 1)/32 is prime.

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A127936 Numbers k such that 1 + Sum_{i=1..k} 2^(2*i-1) is prime.

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A127955 Composite numbers of the form (2^p+1)/3 where p is a prime.

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A127957 Numbers k such that (2^prime(k) + 1)/3 is composite.

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Mathematica

PARI

A127956 Prime numbers p such that (2^p+1)/3 is composite.

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Mathematica

A229145 Numbers k such that (36^k + 1)/37 is prime.

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