A000979 -id:A000979 - cp's OEIS frontend

A123176 Numbers n such that (2^p + 1)/3 is prime, where p is the n-th prime.

2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, 46, 65, 69, 126, 267, 380, 495, 762, 1285, 1304, 1364, 1479, 1697, 4469, 8135, 9193, 11065, 11902, 12923, 13103, 23396, 23642, 31850, 77509, 285228

Offset: 1

Alexander Adamchuk, Oct 03 2006

Also prime(a(n)) are the indices of prime Jacobsthal numbers (A001045) with prime indices. Primes in the Jacobsthal sequence are listed in A049883.

C. Caldwell's The Top Twenty, Wagstaff.

Cf. A000978, A000979, A001045, A049883, A107036, A123214.

Select[Range[500],PrimeQ[(2^Prime[#]+1)/3]&] (* The program generates the first 23 terms of the sequence. To generate more, increase the Range constant. *) (* Harvey P. Dale, Mar 09 2022 *)

a(n) = A000720( A000978(n) ).

Two more terms computed from A000978 by Max Alekseyev, Mar 03 2010

A123214 Primes q such that (2^p + 1)/3 is prime, where p = Prime[q]; or primes in A123176[n].

Original entry on oeis.org

2, 3, 5, 7, 11, 31, 43, 1697, 12923, 13103, 77509

Offset: 1

Alexander Adamchuk, Oct 05 2006

A123176[n] are the numbers n such that (2^p + 1)/3 is prime, where p = Prime[n]. A123176[n] = PrimePi[A000978[n]]. PrimePi[a(n)] = {1,2,3,4,5,11,14,265,1540,1559,...}.

			A123176[n] begin {2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, 31, 39, 43, ...}.
Thus
a(1) = 2, a(2) = 3, a(3) = 5, a(4) = 7, a(5) = 11, a(6) = 31, a(7) = 43.

Cf. A123176, A000978, A000979, A001045, A049883, A107036.

One more term from Max Alekseyev, Feb 06 2010

A246758 Prime numbers of the form (2^(mn)-1)/((2^m-1)(2^n-1)).

Original entry on oeis.org

3, 11, 43, 151, 683, 2731, 43691, 174763, 599479, 2796203, 715827883, 2932031007403, 10052678938039, 145295143558111, 581283643249112959, 658812288653553079, 768614336404564651, 9520972806333758431, 201487636602438195784363

Offset: 1

Nico Brown, Sep 02 2014

nonn

The sequence contains A000979 as a subsequence.

Both m and n must be prime.

			For m=3 and n=5, (2^15-1)/((2^3-1)(2^5-1))=151 is prime, so 151 is a member of the sequence.

Robert Israel, Table of n, a(n) for n = 1..39

Primes in A140803.

N:= 200: # to use all (p, q) with p*q < N
Primes:= select(isprime, [$2..floor(N/2)]):
A:= {}:
for i from 1 to nops(Primes) do
  p:= Primes[i];
  Qs:= select(q -> q < N/p, [seq(Primes[j], j=1..i-1)]);
  A:= A union {seq((2^(p*q)-1)/(2^p-1)/(2^q-1), q=Qs)};
od:
# in Maple 12 and up
select(isprime, A);
# or in earlier Maple versions
sort([select(isprime, , A); # _)[]])[];
# Robert Israel, Sep 02 2014

A302333 Wagstaff primes related to The New Mersenne Conjecture that are the indices of perfect numbers in a list of centered 9-gonal numbers.

Original entry on oeis.org

3, 11, 43, 2731, 43691, 174763, 715827883, 768614336404564651, 56713727820156410577229101238628035243

Offset: 1

Steve Homewood, Apr 05 2018

Let p be a Wagstaff prime related to The New Mersenne Conjecture. Then (3p-2)(3p-1)/2 gives the perfect number whose index it is.

			For p = 3, (3*3-2)*(3*3-1)/2 = 28 and for p = 11, (3*11-2)(3*11-1)/2 = 496.

Caldwell and Honaker, 56713...35243 (38-digits), Prime Curios!
Eric Weisstein's World of Mathematics, New Mersenne Prime Conjecture

Cf. A000979, A107360.

A127959 Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).

Original entry on oeis.org

171, 10923, 699051, 11184811, 44739243, 178956971, 2863311531, 11453246123, 45812984491, 183251937963, 733007751851, 11728124029611, 46912496118443, 187649984473771, 750599937895083, 3002399751580331, 12009599006321323

Offset: 1

Artur Jasinski, Feb 09 2007

nonn

Prime numbers of the form 1 + Sum_{k=1..m} 2^(2*n - 1) is A000979. Numbers x such that 1 + Sum_{k=1..m} 2^(2*n - 1) is prime for n=1,2,...,x is A127936. A127955 is probably a subset of the present sequence.

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

a = {}; Do[c = 1 + Sum[2^(2n - 1), {n, 1, x}]; If[PrimeQ[c] == False, AppendTo[a, c]], {x, 1, 50}]; a
Select[Table[Sum[2^(2k-1),{k,n}]+1,{n,50}],!PrimeQ[#]&] (* Harvey P. Dale, Dec 23 2017 *)

A185156 Primes with the property that complementing any two different bits in the binary representation of these primes never produces a prime number.

Original entry on oeis.org

2, 3, 2731, 174763, 715827883, 1464948053

Offset: 1

Terentyev Oleg, Dec 22 2011

Also called weakly primes of 2nd order in base 2.

Formal definition: let P = set of prime numbers, XOR(x,y) = bitwise x xor y, set of witnesses for an integer x>1 w(x) := Union_{1<=k<=floor(log_2(x)), 0<=j

There are only 6 terms < 10^11 (exhaustive search). But several larger terms of a special form are known (Wagstaff primes, A000979). The smallest of them are:

a(6+)=2932031007403,

a(7+)=768614336404564651,

a(8+)=201487636602438195784363. - Terentyev Oleg

			a(3)=2731 is in the sequence because it is prime and all its witnesses are composite numbers :
2731  =  101010101011 ->       10101011  =     171  =  3^2 * 19
                             1000101011  =     555  =  3 * 5 * 37
                             1010001011  =     651  =  3 * 7 * 31
                             1010100011  =     675  =  3^3 * 5^2
                             1010101001  =     681  =  3 * 227
                             1010101010  =     682  =  2 * 11 * 31
                             1010101111  =     687  =  3 * 229
                             1010111011  =     699  =  3 * 233
                             1011101011  =     747  =  3^2 * 83
                             1110101011  =     939  =  3 * 313
                            11010101011  =    1707  =  3 * 569
                           100000101011  =    2091  =  3 * 17 * 41
                           100010001011  =    2187  =  3^7
                           100010100011  =    2211  =  3 * 11 * 67
                           100010101001  =    2217  =  3 * 739
                           100010101010  =    2218  =  2 * 1109
                           100010101111  =    2223  =  3^2 * 13 * 19
                           100010111011  =    2235  =  3 * 5 * 149
                           100011101011  =    2283  =  3 * 761
                           100110101011  =    2475  =  3^2 * 5^2 * 11
                           101000001011  =    2571  =  3 * 857
                           101000100011  =    2595  =  3 * 5 * 173
                           101000101001  =    2601  =  3^2 * 17^2
                           101000101010  =    2602  =  2 * 1301
                           101000101111  =    2607  =  3 * 11 * 79
                           101000111011  =    2619  =  3^3 * 97
                           101001101011  =    2667  =  3 * 7 * 127
                           101010000011  =    2691  =  3^2 * 13 * 23
                           101010001001  =    2697  =  3 * 29 * 31
                           101010001010  =    2698  =  2 * 19 * 71
                           101010001111  =    2703  =  3 * 17 * 53
                           101010011011  =    2715  =  3 * 5 * 181
                           101010100001  =    2721  =  3 * 907
                           101010100010  =    2722  =  2 * 1361
                           101010100111  =    2727  =  3^3 * 101
                           101010101000  =    2728  =  2^3 * 11 * 31
                           101010101101  =    2733  =  3 * 911
                           101010101110  =    2734  =  2 * 1367
                           101010110011  =    2739  =  3 * 11 * 83
                           101010111001  =    2745  =  3^2 * 5 * 61
                           101010111010  =    2746  =  2 * 1373
                           101010111111  =    2751  =  3 * 7 * 131
                           101011001011  =    2763  =  3^2 * 307
                           101011100011  =    2787  =  3 * 929
                           101011101001  =    2793  =  3 * 7^2 * 19
                           101011101010  =    2794  =  2 * 11 * 127
                           101011101111  =    2799  =  3^2 * 311
                           101011111011  =    2811  =  3 * 937
                           101100101011  =    2859  =  3 * 953
                           101110001011  =    2955  =  3 * 5 * 197
                           101110100011  =    2979  =  3^2 * 331
                           101110101001  =    2985  =  3 * 5 * 199
                           101110101010  =    2986  =  2 * 1493
                           101110101111  =    2991  =  3 * 997
                           101110111011  =    3003  =  3 * 7 * 11 * 13
                           101111101011  =    3051  =  3^3 * 113
                           110010101011  =    3243  =  3 * 23 * 47
                           111000101011  =    3627  =  3^2 * 13 * 31
                           111010001011  =    3723  =  3 * 17 * 73
                           111010100011  =    3747  =  3 * 1249
                           111010101001  =    3753  =  3^3 * 139
                           111010101010  =    3754  =  2 * 1877
                           111010101111  =    3759  =  3 * 7 * 179
                           111010111011  =    3771  =  3^2 * 419
                           111011101011  =    3819  =  3 * 19 * 67
                           111110101011  =    4011  =  3 * 7 * 191

isWPof2ndOrderBase2[x_] := Module[{j = 1, k = 2, flag = x <= 3 || ! BitAnd[x - 3, x - 4] == 0, bitlen = BitLength@x}, While[flag && k < bitlen, While[flag && j < k, flag = !PrimeQ@BitXor[x, BitShiftLeft[1, j] + BitShiftLeft[1, k]]; j++]; j = 1; k++]; flag]; Select[Prime[Range[20000]], isWPof2ndOrderBase2]

A243979 Indices of Wagstaff primes.

Original entry on oeis.org

2, 5, 14, 124, 399, 4552, 15898, 203095, 37029521, 105973558438, 19140185454656173, 3827634977577891833517

Offset: 1

Omar E. Pol, Jun 18 2014

			For n = 3 the third Wagstaff prime is A000979(3) = 43 and 43 is also the 14th prime number, so a(3) = 14.

Andrew R. Booker, The Nth Prime Page.
Chris K. Caldwell, Wagstaff, The Top Twenty, The PrimePages.
Xavier Gourdon and Pascal Sebah, Counting primes.
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x).
Samuel S. Wagstaff, Jr., The Cunningham Project.
Kim Walisch, Fast C++ prime counting function implementation (primecount).
Wikipedia, Wagstaff prime.

Cf. A000040, A000720, A000978, A000979, A059305, A123176, A126614, A194810.

default(primelimit, 10^9); forprime(p=3, 31, q=(2^p+1)/3; if(isprime(q), print1(primepi(q)", "))) \\ Jens Kruse Andersen, Jun 22 2014

a(n) = A000720(A000979(n)).

A000040(a(n)) = A000979(n).

a(11) from Jens Kruse Andersen, Jun 22 2014

a(12) calculated using Kim Walisch's primecount and added by Amiram Eldar, Sep 05 2024

A360475 Smallest prime factor of (2^prime(n) + 1) / 3.

Original entry on oeis.org

3, 11, 43, 683, 2731, 43691, 174763, 2796203, 59, 715827883, 1777, 83, 2932031007403, 283, 107, 2833, 768614336404564651, 7327657, 56409643, 1753, 201487636602438195784363, 499, 179, 971, 845100400152152934331135470251, 415141630193, 643, 104124649, 227

Offset: 2

Alain Rocchelli, Feb 08 2023

nonn

If (2^prime(n) + 1) / 3 is prime then a(n) is a Wagstaff prime (cf. A000979).

For n > 2, a(n) is congruent to 1 (mod 2*prime(n)).

			a(2)=3 since for prime(2)=3, (2^3+1)/3 = 3;
a(3)=11 since for prime(3)=5, (2^5+1)/3 = 11;
a(10)=59 since for prime(10)=29, (2^29+1)/3 = 59*3033169.

Cf. A020639, A126614.

a:= n-> min(numtheory[factorset]((2^ithprime(n)+1)/3)):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 28 2023

a[n_] := FactorInteger[(2^Prime[n]+1)/3][[1, 1]];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 27 2025 *)

forprime(p=3, 100, An=(2^p+1)/3; if(isprime(An), print1(An,", "), forprime(div=3, 2^((p-1)/2), if(An%div==0, print1(div,", "); next(2)))))

a(n) = A020639(A126614(n)).

a(26)-a(30) from Amiram Eldar, Feb 08 2023

Previous Showing 21-28 of 28 results.

cp's OEIS Frontend

A123176 Numbers n such that (2^p + 1)/3 is prime, where p is the n-th prime.

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A123214 Primes q such that (2^p + 1)/3 is prime, where p = Prime[q]; or primes in A123176[n].

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A246758 Prime numbers of the form (2^(m*n)-1)/((2^m-1)*(2^n-1)).

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A302333 Wagstaff primes related to The New Mersenne Conjecture that are the indices of perfect numbers in a list of centered 9-gonal numbers.

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A127959 Nonprime numbers of the form 1 + Sum_{k=1..m} 2^(2*k - 1).

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A185156 Primes with the property that complementing any two different bits in the binary representation of these primes never produces a prime number.

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A243979 Indices of Wagstaff primes.

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A360475 Smallest prime factor of (2^prime(n) + 1) / 3.

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A246758 Prime numbers of the form (2^(mn)-1)/((2^m-1)(2^n-1)).