A126614 -id:A126614 - cp's OEIS frontend

A000979 Wagstaff primes: primes of form (2^p + 1)/3.

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243, 62357403192785191176690552862561408838653121833643

Offset: 1

N. J. A. Sloane

nonn

Also, the primes with prime indices in the Jacobsthal sequence A001045.
Indices n such that (2^n + 1)/3 is prime are listed in A000978. - Alexander Adamchuk, Oct 03 2006
Primes in A126614. - Omar E. Pol, Nov 05 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..20
P. Berrizbeitia, F. Luca, and R. Melham, On a compositeness test for (2^p+1)/3, JIS 13 (2010) 10.1.7
C. Caldwell's The Top Twenty, Wagstaff.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer, Math. Mag., 27 (1954), 156-157.
Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer (annotated and scanned copy)
Djurre G. Sikkema, Probable primality testing for Wagstaff prime, Bachelor's project mathematics, Univ. Groningen (Netherlands 2024). See p. 32.
S. S. Wagstaff, Jr., The Cunningham Project.
Wikipedia, Wagstaff prime

Cf. A000978, A049883, A001045, A127962.

Cf. A010051; subsequence of A007583.

a000979 n = a000979_list !! (n-1)
a000979_list = filter ((== 1) . a010051) a007583_list
-- Reinhard Zumkeller, Mar 24 2013

Select[ Array[(2^# + 1)/3 &, 190], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 03 2010 *)

forprime(p=2,10000,if(ispseudoprime(2^p\/3),print1(2^p\/3,","))) \\ Edward Jiang, Sep 05 2014

from gmpy2 import divexact
from sympy import prime, isprime
A000979 = [p for p in (divexact(2**prime(n)+1,3) for n in range(2,10**2)) if isprime(p)] # Chai Wah Wu, Sep 04 2014

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 *)

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

A127962 Binary expansion of A000979(n).

Original entry on oeis.org

11, 1011, 101011, 1010101011, 101010101011, 1010101010101011, 101010101010101011, 1010101010101010101011, 101010101010101010101010101011, 101010101010101010101010101010101010101011

Offset: 1

Artur Jasinski, Feb 09 2007

Amiram Eldar, Table of n, a(n) for n = 1..20

Cf. A000978, A000979, A007088, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961.

b[n_] := FromDigits[IntegerDigits[n, 2]]; b /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[15]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

from gmpy2 import divexact
from sympy import prime, isprime
A127962 = [int(bin(p)[2:]) for p in (divexact(2**prime(n)+1,3) for n in range(2,10**2)) if isprime(p)] # Chai Wah Wu, Sep 04 2014

a(n) = A007088(A000979(n)). - Amiram Eldar, Jul 23 2023

Edited by N. J. A. Sloane, Jun 11 2007

A127963 Number of 1's in A127962(n).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 10, 12, 16, 22, 31, 40, 51, 64, 84, 96, 100, 157, 174, 351, 855, 1309, 1770, 2904, 5251, 5346, 5640, 6196, 7240, 21369, 41670, 47685, 58620, 63516, 69469, 70540, 133509, 134994, 187161, 493096, 2015700

Offset: 1

Artur Jasinski, Feb 09 2007

Cf. A000120, A000978, A000979, A007953, A124400, A126614, A127936, A127955, A127956, A127957, A127958, A127961, A127962.

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, DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d (* Artur Jasinski, Feb 09 2007 *)
DigitCount[#, 2, 1]& /@ Select[Table[(2^p + 1)/3, {p, Prime[Range[300]]}], PrimeQ] (* Amiram Eldar, Jul 23 2023 *)

a(n) = A000120(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = A007953(A127962(n)). - Amiram Eldar, Jul 23 2023

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Jul 23 2023

A127964 Number of 0's in the binary expansion of A127962(n).

Original entry on oeis.org

0, 1, 2, 4, 5, 7, 8, 10, 14, 20, 29, 38, 49, 62, 82, 94, 98, 155, 172, 349, 853, 1307, 1768, 2902, 5249, 5344, 5638, 6194, 7238, 21367, 41668, 47683, 58618, 63514, 69467, 70538, 133507, 134992, 187159, 493094, 2015698

Offset: 1

Artur Jasinski, Feb 09 2007

Apparently numbers k such that (2^(2*k+3)+1)/3 is prime. - James R. Buddenhagen, Apr 14 2011 [This is true. See the second formula. - Amiram Eldar, Oct 13 2024]

Cf. A000978, A000979, A023416, A127962, A127961, A000979, A000978, A124400, A126614, A127955, A127956, A127957, A127958, A127936.

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, DigitCount[a[[x]], 10, 0]], {x, 1, Length[a]}]; d
(Select[Prime[Range[200]], PrimeQ[(2^# + 1)/3] &] - 3)/2 (* Amiram Eldar, Oct 13 2024 *)

a(n) = A023416(A000979(n)). - Michel Marcus, Nov 07 2013

a(n) = (A000978(n)-3)/2. - Amiram Eldar, Oct 13 2024

a(22)-a(29) from Vincenzo Librandi, Mar 31 2012

a(30)-a(41) from Amiram Eldar, Oct 13 2024

A127965 Number of bits in A127962(n).

Original entry on oeis.org

2, 4, 6, 10, 12, 16, 18, 22, 30, 42, 60, 78, 100, 126, 166, 190, 198, 312, 346, 700, 1708, 2616, 3538, 5806, 10500, 10690, 11278, 12390, 14478, 42736, 83338, 95368, 117238, 127030, 138936, 141078, 267016, 269986, 374320, 986190, 4031398

Offset: 1

Artur Jasinski, Feb 09 2007

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

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, DigitCount[a[[x]], 10, 0]+DigitCount[a[[x]], 10, 1]], {x, 1, Length[a]}]; d

a(n) = A127964(n) + A127963(n).

a(n) = 1 + floor(log_2(A000979(n))) = 1 + floor(log_2(2^A000978(n)+1) - A020857) = A000978(n) - 1. - R. J. Mathar, Feb 01 2008

a(22)-a(29) from Vincenzo Librandi, Mar 30 2012

a(30)-a(41) from Amiram Eldar, Oct 19 2024

cp's OEIS Frontend

A000979 Wagstaff primes: primes of form (2^p + 1)/3.

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

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

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A127962 Binary expansion of A000979(n).

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A127963 Number of 1's in A127962(n).

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