A000978 -id:A000978 - cp's OEIS frontend

A359436 Primes p such that (4^p - 2^p + 1)/3 is prime.

3, 5, 7, 13, 29, 61, 383, 401, 1637, 1871, 36229, 44771, 44797, 75167

Offset: 1

Jorge Coveiro, Dec 31 2022

Terms > 1871 correspond to probable primes.

Is 9 the only composite k such that (4^k - 2^k + 1)/3 is prime? Checked up to 20000. - Andrew Howroyd, Sep 10 2024

			3 is a term because 3 is prime and (4^3 - 2^3 + 1)/3 = 19 is also prime.

Cf. A000978.

isok(k)={k%2 && ispseudoprime((4^k - 2^k + 1)/3)}
{ forprime(p=3, 2000, if(isok(p), print1(p, ", "))) } \\ Andrew Howroyd, Dec 31 2022

A361562 Wagstaff numbers that are of the form 4*k + 3.

Original entry on oeis.org

3, 7, 11, 19, 23, 31, 43, 79, 127, 167, 191, 199, 347, 3539, 5807, 10691, 11279, 12391, 14479, 83339, 117239, 127031, 141079, 269987, 986191, 4031399

Offset: 1

Jorge Coveiro, Mar 15 2023

13347311 and 13372531 are also in the sequence, but may not be the next terms.

Jorge Coveiro, Possible 'Formula' for Wagstaff numbers, mersenneforum.org.

Cf. A000978 (Wagstaff numbers), A002145 (primes of form 4*k+3), A112633, A361563.

from itertools import count, islice
from sympy import prime, isprime
def A361562_gen(): # generator of terms
    return filter(lambda p: p&2 and isprime(((1<A361562_list = list(islice(A361562_gen(),10)) # Chai Wah Wu, Mar 21 2023

Intersection of A000978 and A002145.

A361563 Wagstaff numbers that are of the form 4*k + 1.

Original entry on oeis.org

5, 13, 17, 61, 101, 313, 701, 1709, 2617, 10501, 42737, 95369, 138937, 267017, 374321

Offset: 1

Jorge Coveiro, Mar 15 2023

15135397 is also in the sequence, but may not be the next term.

Jorge Coveiro, Possible 'Formula' for Wagstaff numbers, mersenneforum.org.

Cf. A000978 (Wagstaff numbers), A002144 (primes of form 4*k + 1), A112634, A361562.

from itertools import count, islice
from sympy import prime, isprime
def A361563_gen(): # generator of terms
    return filter(lambda p: not p&2 and isprime(((1<A361563_list = list(islice(A361563_gen(),7)) # Chai Wah Wu, Mar 21 2023

Intersection of A000978 and A002144.

A109138 Numbers n such that 2^n + 2 is an admirable number (A111592).

Original entry on oeis.org

6, 8, 12, 14, 18, 20, 24, 32, 44, 62, 80, 102, 128, 168, 192, 200, 314, 348, 702

Offset: 1

Jason Earls, Aug 18 2005

For k > 1, A000978(k)+1 is a member. Are there any others? - David Wasserman, May 28 2008

No more terms below 1064. - Amiram Eldar, Oct 12 2019

			a(3)=12 because 2^12 + 2 = 4098 and 1+2+3+683+1366+2049-6 = 4098.

Cf. A000978.

fQ[n_] := Block[{d = Most[ Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Do[ If[ fQ[2^n + 2], Print[n]], {n, 200}] (* Robert G. Wilson v, Aug 30 2005 *)

More terms from David Wasserman, May 28 2008

a(19) from Amiram Eldar, Oct 12 2019

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 *)

A185343 Least positive number k such that k*p+1 divides 2^p+1 where p is prime(n), or 0 if no such number exists.

Original entry on oeis.org

2, 0, 2, 6, 62, 210, 2570, 9198, 121574, 2, 23091222, 48, 2, 68186767614, 6, 2, 48, 12600235023025650, 109368, 794502, 24, 2550476412689091085878, 6, 2, 10, 8367330694575771627040945250, 4030501264, 6, 955272, 2, 446564785985483547852197647548252246, 8, 8, 32424, 8

Offset: 1

Bill McEachen, Feb 26 2011

Akin to A186283 except for 2^p+1 and restricted to primes.
The larger terms of this sequence occur for the primes p > 3 in sequence A000978. These large terms are (2^p-2)/(3p).
a(n) = 2 iff prime(n) is in A103579. - Robert Israel, Jul 17 2023

			2^3+1 = 9 has no factor of the form k*3+1 except 1, so a(primepi(3)) = a(2) = 0.
2^29+1 = 536870913 has factor 2*29+1=59, so a(primepi(29)) = a(10) = 2.

Cf. A098268, A103579, A186283.

f:= proc(n) local p,F;
  p:= ithprime(n);
  F:= select(t -> t mod p = 1, numtheory:-divisors(2^p+1) minus {1});
  if F = {} then 0 else (min(F)-1)/p; fi
end proc:
map(f, [$1..50]); # Robert Israel, Jul 17 2023

Table[q = First /@ FactorInteger[2^p + 1]; s = Select[q, Mod[#1, p] == 1 &, 1]; If[s == {}, 0, (s[[1]] - 1)/p], {p, Prime[Range[30]]}]

A215937 Numbers n such that 2^n + 1 can be written in the form a^2 + 5*b^2.

Original entry on oeis.org

2, 3, 7, 10, 11, 19, 23, 31, 43, 47, 50, 58, 71, 79, 82, 107, 127, 167, 178, 179, 191, 199, 250, 290, 298, 311, 347, 359, 410, 487, 563, 599, 683, 751, 802, 890, 907, 1051

Offset: 1

V. Raman, Aug 27 2012

nonn

These 2^n + 1 numbers can only have prime factors of the form 1 (mod 20) or 3 (mod 20) or 5 (mod 20) or 7 (mod 20) or 9 (mod 20) raised to an odd power, but their overall product 2^n+1 can only be 1 (mod 20) or 5 (mod 20) or 9 (mod 20). This statement is limited to odd numbers.
In general,
A number n can be written in the form a^2+5*b^2 if and only if n is 0,
or of the form 2^(2i) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m)
or of the form 2^(2i+1) 5^j Prod_{p==1 or 9 mod 20} p^k Prod_{q==3 or 7 mod 20) q^(2m+1),
for integers i,j,k,m, for primes p,q.

			3 is in the sequence because 2^3 + 1 = 9 can be written as 2^2 + 5 * 1^2 = 9.

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

Cf. A000051, A000978, A215800, A215801.

Cf. A020669, A033205 (numbers and primes of the form x^2 + 5*y^2).

for(i=2, 500, a=factorint(2^i+1)~; has=0; for(j=1, #a, if(((a[1, j]%20>10)||(i%4<2))&&a[2, j]%2==1, has=1; break)); if(has==0, print(i",")))

for(i=2, 500, a=factorint(2^i+1)~; flag=0; flip=0; for(j=1, #a, if(((a[1, j]%20>10))&&a[2, j]%2==1, flag=1); if(((a[1, j]%20==2)||(a[1, j]%20==3)||(a[1, j]%20==7))&&a[2, j]%2==1, flip=flip+1)); if(flag==0&&flip%2==0, print(i",")))

Terms corrected by V. Raman, Sep 20 2012

A216550 Numbers n such that (2^n+1)/3 is prime and can be written in the form a^2 + 3*b^2.

Original entry on oeis.org

7, 13, 19, 31, 43, 61, 79, 127, 199, 313, 2617, 10501, 12391, 14479, 138937, 141079, 986191

Offset: 1

V. Raman, Sep 08 2012

nonn

The exponent is congruent to 1 mod 3.

Samuel S. Wagstaff, Jr. The Cunningham Project

Cf. A000978.

A216551 Numbers n such that (2^n+1)/3 is prime, but cannot be written in the form a^2 + 3*b^2.

Original entry on oeis.org

5, 11, 17, 23, 101, 167, 191, 347, 701, 1709, 3539, 5807, 10691, 11279, 42737, 83339, 95369, 117239, 127031, 267017, 269987, 374321, 4031399

Offset: 1

V. Raman, Sep 08 2012

nonn

The exponent is congruent to 2 mod 3.

Samuel S. Wagstaff, Jr. The Cunningham Project

Cf. A000978.

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

cp's OEIS Frontend

A359436 Primes p such that (4^p - 2^p + 1)/3 is prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A361562 Wagstaff numbers that are of the form 4*k + 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A361563 Wagstaff numbers that are of the form 4*k + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Python

Formula

A109138 Numbers n such that 2^n + 2 is an admirable number (A111592).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A185343 Least positive number k such that k*p+1 divides 2^p+1 where p is prime(n), or 0 if no such number exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A215937 Numbers n such that 2^n + 1 can be written in the form a^2 + 5*b^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Extensions

A216550 Numbers n such that (2^n+1)/3 is prime and can be written in the form a^2 + 3*b^2.

Views

Author

Keywords

Comments

Links

Crossrefs

A216551 Numbers n such that (2^n+1)/3 is prime, but cannot be written in the form a^2 + 3*b^2.

Views

Author