A127578 -id:A127578 - cp's OEIS frontend

A035050 a(n) is the smallest k such that k*2^n + 1 is prime.

1, 1, 1, 2, 1, 3, 3, 2, 1, 15, 12, 6, 3, 5, 4, 2, 1, 6, 3, 11, 7, 11, 25, 20, 10, 5, 7, 15, 12, 6, 3, 35, 18, 9, 12, 6, 3, 15, 10, 5, 6, 3, 9, 9, 15, 35, 19, 27, 15, 14, 7, 14, 7, 20, 10, 5, 27, 29, 54, 27, 31, 36, 18, 9, 12, 6, 3, 9, 31, 23, 39, 39, 40, 20, 10, 5, 58, 29, 15, 36, 18, 9, 13

Offset: 0

Labos Elemer

nonn

From Ulrich Krug (leuchtfeuer37(AT)gmx.de), Jun 05 2010: (Start)
If a(i) = 2 * m then a(i+1) = m.
Proof: (I) a(i) = 2*m, 2*m * 2^i + 1 = m*2^(i+1) + 1 prime, so a(i+1) <= m;
(II) if a(i+1) = m-d for an integer d > 0, (m-d) * 2^(i+1) + 1 = (2*m-2*d) * 2^i + 1 prime;
(2m-2d) < 2m contradiction to a(i) = 2 * m, d = 0.
(End)
Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - Thomas Ordowski, Apr 13 2019

			a(3)=2 because 1*2^3 + 1 = 9 is composite, 2*2^3 + 1 = 17 is prime.
a(99)=219 because 2^99k + 1 is not prime for k=1,2,...,218. The first term which is not a composite number of this arithmetic progression is 2^99*219 + 1.

T. D. Noe, 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.
Index entries for sequences related to primes in arithmetic progressions

Analogous case is A034693. Special subscripts (n's for a(n)=1) are the exponents of known Fermat primes: A000215. See also Fermat numbers A000051.

Cf. A007522, A057778, A080076, A085427, A087522, A126717, A127575, A127576, A127577, A127578, A127580, A127581, A127586.

sol:=[];m:=1; for n in [0..82] do k:=0; while not IsPrime(k*2^n+1) do k:=k+1; end while; sol[m]:=k; m:=m+1; end for; sol; // Marius A. Burtea, Jun 05 2019

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 1], k++ ]; AppendTo[a, k], {n, 0, 100}]; a (* Artur Jasinski *)
Table[Module[{k=1,n2=2^n},While[!PrimeQ[k*n2+1],k++];k],{n,0,90}] (* Harvey P. Dale, May 25 2024 *)

a(n) = {my(k = 1); while (! isprime(2^n*k+1), k++); k;}

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

A127580 Numbers k such that 64k+63 is prime.

Original entry on oeis.org

1, 2, 5, 16, 17, 19, 22, 25, 32, 34, 41, 44, 52, 55, 61, 64, 74, 79, 85, 94, 95, 97, 104, 107, 109, 110, 116, 127, 131, 137, 142, 145, 152, 157, 160, 164, 166, 172, 179, 184, 185, 194, 197, 199

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A127575, A127576, A127577, A127578, A127580, A127581.

a = {}; Do[If[PrimeQ[64n + 63], AppendTo[a, n]], {n, 1, 200}]; a

A127577 Numbers n for which 32n+31 is prime.

Original entry on oeis.org

3, 5, 6, 11, 14, 18, 26, 30, 33, 35, 39, 44, 45, 48, 51, 54, 56, 60, 65, 66, 68, 69, 74, 80, 83, 84, 89, 98, 104, 105, 111, 123, 128, 129, 138, 144, 146, 149, 150, 156, 158, 159, 164, 168, 170, 171, 180, 188, 189, 191, 195, 198

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Cf. A127575, A127576, A127578, A127579, A127580, A127581.

a = {}; Do[If[PrimeQ[32n + 31], AppendTo[a, n]], {n, 1, 200}]; a
Select[Range[200],PrimeQ[32#+31]&] (* Harvey P. Dale, Dec 14 2024 *)

A127586 Smallest strictly positive integer k such that (k+1)*2^n-1 is prime.

Original entry on oeis.org

2, 1, 1, 2, 1, 3, 1, 2, 4, 6, 4, 2, 1, 8, 4, 3, 1, 3, 1, 27, 13, 6, 25, 12, 38, 21, 10, 15, 7, 3, 1, 9, 4, 5, 2, 23, 11, 5, 2, 24, 23, 11, 5, 2, 13, 6, 19, 9, 4, 18, 10

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

The associated prime number list is (k+1)*2^n-1 = 2, 3, 7, 23, 31, 127, 127, 383, 1279, 3583, 5119, 6143, 8191, 73727 for n=0,1,2,3,4,... - R. J. Mathar, Jan 22 2007

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

A127586 := proc(n) local k; k:=1 ; while true do if isprime( (k+1)*2^n-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 0 to 100 do printf("%d, ",A127586(n)) ; od ; # R. J. Mathar, Jan 22 2007

a = {}; Do[k = 1; While[ ! PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k], {n, 0, 50}]; a

a(n)=A127587(n) if n is not in A000043. - R. J. Mathar, Jan 22 2007

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

A127587 Smallest nonnegative integer k such that (k+1)*2^n-1 is prime.

Original entry on oeis.org

2, 1, 0, 0, 1, 0, 1, 0, 4, 6, 4, 2, 1, 0, 4, 3, 1, 0, 1, 0, 13, 6, 25, 12, 38, 21, 10, 15, 7, 3, 1, 0, 4, 5, 2, 23, 11, 5, 2, 24, 23, 11, 5, 2, 13, 6, 19, 9, 4, 18, 10, 20, 19, 9, 4, 2, 31, 15, 7, 3, 1, 0, 11, 5, 2, 66, 62, 42, 62, 39, 19, 9, 4, 14, 11, 5, 2, 54, 46, 29, 14, 29, 14, 63, 31, 15, 7

Offset: 0

Artur Jasinski, Jan 19 2007

nonn

The associated prime number list is (k+1)*2^n-1 = 2,3,3,7,31,31,127,127,1279,3583,5119,6143,... for n=0,1,2,3,4,... - R. J. Mathar, Jan 22 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

A127587 := proc(n) local k; k:=0 ; while true do if isprime( (k+1)*2^n-1) then RETURN(k) ; fi ; k := k+1 ; od ; end: for n from 0 to 100 do printf("%d, ",A127587(n)) ; od ; # R. J. Mathar, Jan 22 2007

a = {}; Do[k = 0; While[ ! PrimeQ[k 2^n + 2^n - 1], k++ ]; AppendTo[a, k], {n, 0, 50}]; a

a[A000043(j)]=0 for j=1,2,3,4,... - R. J. Mathar, Jan 22 2007

a(n) = A085427(n) - 1. - Filip Zaludek, Dec 16 2016

More terms from R. J. Mathar, Jan 22 2007

A127589 Primes of the form 16k + 5.

Original entry on oeis.org

5, 37, 53, 101, 149, 181, 197, 229, 277, 293, 373, 389, 421, 613, 661, 677, 709, 757, 773, 821, 853, 997, 1013, 1061, 1093, 1109, 1237, 1301, 1381, 1429, 1493, 1621, 1637, 1669, 1733, 1861, 1877, 1973, 2053, 2069, 2213, 2293, 2309, 2341, 2357, 2389, 2437

Offset: 1

Artur Jasinski, Jan 19 2007

All terms are the sum of two squares.

Primes with least significant digit 5 in hexadecimal. - Alonso del Arte, Oct 21 2022

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587.

a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, 16n + 5]], {n, 0, 200}]; a
Select[16Range[200] + 5, PrimeQ] (* Alonso del Arte, Oct 21 2022 *)

select(x->(x%16)==5, primes(500)) \\ Michel Marcus, Oct 24 2022

Invalid comment removed by Zak Seidov, Jul 22 2010

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

A127590 Numbers n such that 16n+5 is prime.

Original entry on oeis.org

0, 2, 3, 6, 9, 11, 12, 14, 17, 18, 23, 24, 26, 38, 41, 42, 44, 47, 48, 51, 53, 62, 63, 66, 68, 69, 77, 81, 86, 89, 93, 101, 102, 104, 108, 116, 117, 123, 128, 129, 138, 143, 144, 146, 147, 149, 152, 159, 167, 168, 171, 174, 177, 182, 191, 194

Offset: 1

Artur Jasinski, Jan 19 2007

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A035050, A007522, A127575, A127576, A127577, A127578, A127580, A127581, A087522, A127586, A127587, A127589.

a = {}; Do[If[PrimeQ[16n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Range[0,200],PrimeQ[16#+5]&] (* Harvey P. Dale, Aug 31 2020 *)

is(n)=isprime(16*n+5) \\ Charles R Greathouse IV, Feb 17 2017

A101998 Primes of the form 32k-1 such that 4k-1, 8k-1, 16k-1 and 64*k-1 are also primes.

Original entry on oeis.org

1439, 429119, 507359, 1014719, 1017119, 2034239, 2368799, 2727359, 4858559, 6484319, 8553599, 8981279, 12789599, 12972959, 14567999, 14929919, 15301439, 15367679, 16362719, 17107199, 17263199, 17962559, 18224639, 18857759

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 23 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719, 32*45-1 = 1439 and 64*45-1 = 2879 are primes, so 1439 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A101994, A101995, A101996, A101997, A101999.

Subsequence of A127578 and A101798.

32 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 6]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)

is(k) = if(k % 32 == 31, my(m = k\32 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1) && isprime(64*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 32*A101994(n) - 1 = 8*A101995(n) + 7 = 4*A101996(n) + 3 = 2*A101997(n) + 1. - Amiram Eldar, May 13 2024

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.

Original entry on oeis.org

1439, 2879, 21599, 28319, 96959, 137279, 219839, 429119, 462719, 507359, 571199, 597599, 659999, 700319, 811199, 858239, 861599, 903359, 976799, 982559, 1014719, 1017119, 1067999, 1115519, 1333919, 1342079, 1837919, 2029439, 2034239

Offset: 1

Douglas Stones (dssto1(AT)student.monash.edu.au), Dec 16 2004

			4*45-1 = 179, 8*45-1 = 359, 16*45-1 = 719 and 32*45-1 = 1439 are primes, so 1439 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002515, A101794, A101795, A101796, A101797.
Subsequence of A127578.
Subsequence: A101998.

32 * Select[Range[10^5], And @@ PrimeQ[2^Range[2, 5]*# - 1] &] - 1 (* Amiram Eldar, May 13 2024 *)
Select[Prime[Range[200000]],Mod[#,32]==31&&AllTrue[{4,8,16} (#+1)/32-1,PrimeQ]&] (* Harvey P. Dale, Feb 20 2025 *)

is(k) = if(k % 32 == 31, my(m = k\32 + 1); isprime(4*m-1) && isprime(8*m-1) && isprime(16*m-1) && isprime(32*m-1), 0); \\ Amiram Eldar, May 13 2024

a(n) = 32*A101794(n) - 1 = 8*A101795(n) + 7 = 4*A101796(n) + 3 = 2*A101797(n) + 1. - Amiram Eldar, May 13 2024

cp's OEIS Frontend

A035050 a(n) is the smallest k such that k*2^n + 1 is prime.

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Magma

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A127580 Numbers k such that 64k+63 is prime.

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A127577 Numbers n for which 32n+31 is prime.

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A127586 Smallest strictly positive integer k such that (k+1)*2^n-1 is prime.

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Maple

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A127587 Smallest nonnegative integer k such that (k+1)*2^n-1 is prime.

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Maple

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A127589 Primes of the form 16k + 5.

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

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Mathematica

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A127590 Numbers n such that 16n+5 is prime.

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Mathematica

A101998 Primes of the form 32k-1 such that 4k-1, 8k-1, 16k-1 and 64*k-1 are also primes.

A101798 Primes of the form 32k-1 such that 4k-1, 8k-1 and 16k-1 are also primes.