A057775 -id:A057775 - cp's OEIS frontend

A201364 Numbers k such that A057775(k) is the factor of a Fermat number 2^(2^m) + 1 for some m.

1, 2, 4, 7, 8, 14, 16, 25, 39, 41, 57, 67, 75, 120, 127, 147, 209, 229, 231, 290, 302, 320, 455, 547, 558, 747, 1553, 1947, 2027, 2458, 3313, 3508, 4262, 4727, 6210, 6393, 6539, 6838, 7312, 8242, 8557, 9431, 9450, 12189, 13252, 14254, 14280, 15164, 17909, 18759

Offset: 1

Arkadiusz Wesolowski, Nov 30 2011

nonn

Indices of Fermat factors in A057775.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..57
Wilfrid Keller, Fermat factoring status
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A057775, A057778.

lst = {}; Do[k = 1; While[! PrimeQ[p = (2*k - 1)*2^n + 1], k++]; If[IntegerQ[Log[2, MultiplicativeOrder[2, p]]], AppendTo[lst, n]], {n, 320}]; lst

isok(n)=my(k=-1, p(k)=k*2^n+1, z(k)=znorder(Mod(2, p(k)))); until(isprime(p(k)), k=k+2); z(k)>>valuation(z(k), 2)==1; \\ Arkadiusz Wesolowski, May 26 2023

a(44)-a(50) from Arkadiusz Wesolowski, May 26 2023

A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

41, 209, 157169, 213321, 303093, 382449, 2145353, 2478785

Offset: 1

Arkadiusz Wesolowski, Jan 17 2012

Terms are odd: by Morehead's theorem, 3*2^(2*n) + 1 can never divide a Fermat number.
No other terms below 7516000.
Is this sequence the same as "Numbers k such that 3*2^k + 1 is a factor of a Fermat number 2^(2^m) + 1 for some m"? - Arkadiusz Wesolowski, Nov 13 2018
The last sentence of Morehead's paper is: "It is easy to show that composite numbers of the forms 2^kappa * 3 + 1, 2^kappa * 5 + 1 can not be factors of Fermat's numbers." [a proof is needed]. - Jeppe Stig Nielsen, Jul 23 2019
Any factor of a Fermat number 2^(2^m) + 1 of the form 3*2^k + 1 is prime if k < 2*m + 6. - Arkadiusz Wesolowski, Jun 12 2021
If, for any m >= 0, F(m) = 2^(2^m) + 1 has a prime factor p of the form 3*2^k + 1, then F(m)/p is congruent to 11 mod 30. - Arkadiusz Wesolowski, Jun 13 2021
A number k belongs to this sequence if and only if the order of 2 modulo p is not divisible by 3, where p is a prime of the form 3*2^k + 1 (see Golomb paper). - Arkadiusz Wesolowski, Jun 14 2021

Solomon W. Golomb, Properties of the sequence 3.2^n+1, Math. Comp., 30 (1976), 657-663.
Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002253.

Cf. A000215, A039687, A057775, A057778, A201364, A226366.

lst = {}; Do[p = 3*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 209, 2}]; lst

isok(n) = my(p = 3*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

7, 25, 39, 75, 127, 1947, 3313, 23473, 125413

Offset: 1

Arkadiusz Wesolowski, Jun 05 2013

No other terms below 5330000.

The reason all terms are odd is that if k is even, then 5*2^k + 1 == (-1)*(-1)^k + 1 = (-1)*1 + 1 = 0 (mod 3). So if k is even, then 3 divides 5*2^k + 1, and since 3 divides no other Fermat number than F_0=3 itself, we do not have a Fermat factor. - Jeppe Stig Nielsen, Jul 21 2019

Wilfrid Keller, Fermat factoring status
J. C. Morehead, Note on the factors of Fermat's numbers, Bull. Amer. Math. Soc., Volume 12, Number 9 (1906), pp. 449-451.
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002254.

Cf. A000215, A050526, A057775, A057778, A201364, A204620.

lst = {}; Do[p = 5*2^n + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, n]], {n, 7, 3313, 2}]; lst

isok(n) = my(p = 5*2^n + 1, z = znorder(Mod(2, p))); isprime(p) && ((z >> valuation(z, 2)) == 1); \\ Michel Marcus, Nov 10 2018

A035089 Smallest prime of form 2^n*k + 1.

Original entry on oeis.org

2, 3, 5, 17, 17, 97, 193, 257, 257, 7681, 12289, 12289, 12289, 40961, 65537, 65537, 65537, 786433, 786433, 5767169, 7340033, 23068673, 104857601, 167772161, 167772161, 167772161, 469762049, 2013265921, 3221225473, 3221225473, 3221225473, 75161927681

Offset: 0

Labos Elemer

nonn

a(n) is the smallest prime p such that the multiplicative group modulo p has a subgroup of order 2^n. - Joerg Arndt, Oct 18 2020

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.

Analogous case is A034694. Fermat primes (A019434) are a subset. See also Fermat numbers A000215.

Cf. A007522, A057775, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

a = {}; Do[k = 0; While[ !PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k 2^n + 1], {n, 1, 50}]; a (* Artur Jasinski *)

a(n)=for(k=1,9e99,if(ispseudoprime(k<Charles R Greathouse IV, Jul 06 2011

a(0) from Joerg Arndt, Jul 06 2011

A075978 Positions of check bits in code in A075976.

Original entry on oeis.org

524287, 267912191, 4042816543, 13744315618, 22906842477, 40376162084, 62439401835, 75097753016, 148683059317, 290355594575, 498390060837, 652145353600, 1276493816718, 1985213666020, 2887709403613

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

nonn

J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Transactions on Information Theory, 32:337-348, 1986.

Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075976, A075977, A057775, A075972.

A057776 a(n) is the least number k such that prime(k) - 1 is divisible by 2^(n-1) and the quotient is odd.

Original entry on oeis.org

1, 2, 3, 13, 7, 25, 44, 116, 55, 974, 1581, 2111, 1470, 4289, 10847, 15000, 6543, 91466, 62947, 397907, 498178, 1452314, 6025010, 20197904, 38946356, 9385401, 24843812, 98842359, 166808880, 556542914, 154570517, 3132108468, 7417604438, 3217817383, 47999122016

Offset: 1

Labos Elemer, Nov 02 2000

nonn

			For n = 1, a(1) = 1, prime(a(1)) = prime(1) = 2 and prime(1)-1 = 1 is divisible by 2^(n-1) = 2^0 = 1; moreover 2 is the smallest.
For n = 10, a(10) = 974, the 974th prime is 7681, prime(974) - 1 = 7680 = 512*15, is divisible by 2^9 = 512 and the quotient is 15, and there are no other primes such this below 7681.
A057775(30) = 12348030977; a(30) = 556542914. It means that 12348030977 is the 556542914th prime. A057777(30) = 12348030976; when A057777(30) is divided by 2^29, the quotient is 23 = A057778(30).

Amiram Eldar, Table of n, a(n) for n = 1..72

Cf. A000040, A000720, A006093, A057773, A057775, A057776, A057777, A057778.

a(n) = PrimePi(A057775(n-1)). - Amiram Eldar, Mar 16 2025

a(32)-a(35) from Amiram Eldar, Mar 16 2025

A201914 Least prime p such that p+1 is divisible by 2^n and not by 2^(n+1).

Original entry on oeis.org

2, 5, 3, 7, 47, 31, 191, 127, 1279, 3583, 5119, 6143, 20479, 8191, 81919, 294911, 1114111, 131071, 786431, 524287, 17825791, 14680063, 138412031, 109051903, 654311423, 1912602623, 738197503, 2818572287, 7247757311, 3758096383, 228707008511, 2147483647

Offset: 0

T. D. Noe, Dec 27 2011

nonn

See A126717 for the least k such that k*2^n-1 is prime.

For every n >= 1 there are infinitely many prime numbers p such that p + 1 is divisible by 2^n and not by 2^(n + 1). - Marius A. Burtea, Mar 10 2020

Laurențiu Panaitopol, Alexandru Gica, Arithmetic problems and number theory, Ed. Gil, Zalău, (2006), ch. 13, p. 78, pr. 5 (in Romanian).

Donovan Johnson, Table of n, a(n) for n = 0..1000

Cf. A008864 (primes + 1), A057775 (p-1 case), A126717.

For n>0, sequence is first term of A002144, A007520, A141194, A142041, A142939, ...

a:=[]; for n in [0..31] do k:=1; while not IsPrime(k*2^n-1) do k:=k+2; end while; Append(~a,k*2^n-1); end for; a; // Marius A. Burtea, Mar 10 2020

Table[k = 1; While[p = k*2^n - 1; ! PrimeQ[p], k = k + 2]; p, {n, 0, 40}]

A057777 a(n) is the smallest number such that a(n)+1 is a prime and the largest power of 2 which divides it is 2^n.

Original entry on oeis.org

1, 2, 4, 40, 16, 96, 192, 640, 256, 7680, 13312, 18432, 12288, 40960, 114688, 163840, 65536, 1179648, 786432, 5767168, 7340032, 23068672, 104857600, 377487360, 754974720, 167772160, 469762048, 2013265920, 3489660928, 12348030976, 3221225472, 75161927680, 184683593728

Offset: 0

Labos Elemer, Nov 02 2000

nonn

			The 4th term is 40. It is divisible by 8, 40+1 = 41 is prime. Smaller multiples of 8 are not suitable because, e.g., albeit 8|16 and 16+1 = 17 is a prime, but the largest power of 2 that divides 16 is not 8, it is 16. So 16 is not the 3rd, it is the 4th term here.

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A000040, A006093, A057775, A057776.

a(n) = A057775(n) - 1. - Sean A. Irvine, Jun 27 2022

More terms from Larry Reeves (larryr(AT)acm.org), Nov 03 2000

A137990 Least prime p of the form c*3^n+1 with c not divisible by 3.

Original entry on oeis.org

2, 7, 19, 109, 163, 487, 1459, 17497, 52489, 39367, 472393, 4960117, 5314411, 102036673, 19131877, 57395629, 86093443, 258280327, 3874204891, 23245229341, 90656394427, 585779779369, 251048476873, 9790890598009, 4518872583697

Offset: 0

Andrew V. Sutherland, May 01 2008

a(n) is also the least prime such that 3^(n+1), but not 3^(n+2), divides 2^(a(n)-1)-1.

			a(8)=52489 because 52489=8*3^8+1 is prime and no smaller prime p has p-1 divisible by 3^8 but not 3^9.

Carlos Rivera, Puzzle 1129 OEIS A137990, The Prime Puzzles & Problems Connection.

Cf. A005109, A057775.

A334296 Smallest k such that (2k+1)*2^n+1 is prime.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 1, 2, 0, 7, 6, 4, 1, 2, 3, 2, 0, 4, 1, 5, 3, 5, 12, 22, 22, 2, 3, 7, 6, 11, 1, 17, 21, 4, 37, 29, 1, 7, 7, 2, 13, 1, 4, 4, 7, 17, 9, 13, 7, 11, 3, 8, 3, 25, 24, 2, 13, 14, 49, 13, 15, 26, 52, 4, 12, 4, 1, 4, 15, 11, 19, 19, 63, 11, 33, 2, 46

Offset: 0

Mike Speciner, Apr 21 2020

nonn

A057775 is the corresponding sequence of primes.

			a(0)=a(1)=a(2)=0 because 2^0+1=2, 2^1+1=3, 2^2+1=5 are prime.
a(3)=2 because 2^8+1=9 and 3*2^8+1=25 are not prime, but 5*2^8+1=41 is.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A057778, A057775, A035050, A035089.

f:= proc(n) local t, v, k;
   t:= 2^n; v:= -t+1;
   for k from 0 do
      v:= v+2*t;
      if isprime(v) then return k fi
   od
end proc:
map(f, [$0..100]); # Robert Israel, Jul 14 2020

a[n_] := Block[{k = 0}, While[! PrimeQ[(2 k + 1) 2^n + 1], k++]; k]; Array[a, 77, 0] (* Giovanni Resta, May 08 2020 *)

a(n) = my(k=0); while (!isprime((2*k+1)*2^n+1), k++); k; \\ Michel Marcus, Apr 30 2020

from itertools import count
from sympy import isprime
def pow2p1() : # generates the sequence
  for n in count() :
    for k in count() :
      if isprime(((2*k+1)<

a(n) = (A057778(n)-1)/2.

a(n) = ((A057775(n)-1)/2^n-1)/2.

cp's OEIS Frontend

A201364 Numbers k such that A057775(k) is the factor of a Fermat number 2^(2^m) + 1 for some m.

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A204620 Numbers k such that 3*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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A226366 Numbers k such that 5*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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A035089 Smallest prime of form 2^n*k + 1.

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A075978 Positions of check bits in code in A075976.

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A057776 a(n) is the least number k such that prime(k) - 1 is divisible by 2^(n-1) and the quotient is odd.

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A201914 Least prime p such that p+1 is divisible by 2^n and not by 2^(n+1).

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A057777 a(n) is the smallest number such that a(n)+1 is a prime and the largest power of 2 which divides it is 2^n.

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A137990 Least prime p of the form c*3^n+1 with c not divisible by 3.