A033809 -id:A033809 - cp's OEIS frontend

A029810 Erroneous version of A033809, A046067.

1, 2, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, 2, 1, 1, 4, 2, 5, 4, 1

Offset: 1

dead

A040076 Smallest m >= 0 such that n*2^m + 1 is prime, or -1 if no such m exists.

0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 6, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 8, 3, 1, 2, 1, 0, 2, 5, 1, 0, 1, 0, 2, 1, 2, 0, 583, 1, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 5, 0, 4, 7, 1, 2, 1, 0, 2, 1, 1, 0, 3, 0, 2, 1, 1, 4, 3, 0, 2, 3, 1, 0, 1, 2, 4, 1, 2, 0, 1, 1, 8, 7, 2, 582, 1, 0, 2, 1, 1, 0, 3, 0

Offset: 1

David W. Wilson

Sierpiński showed that a(n) = -1 infinitely often. John Selfridge showed that a(78557) = -1 and it is conjectured that a(n) >= 0 for all n < 78557.

Determining a(131072) = a(2^17) is equivalent to finding the next Fermat prime after F_4 = 2^16 + 1. - Jeppe Stig Nielsen, Jul 27 2019

			1*(2^0)+1=2 is prime, so a(1)=0;
3*(2^1)+1=5 is prime, so a(3)=1;
For n=7, 7+1 and 7*2+1 are composite, but 7*2^2+1=29 is prime, so a(7)=2.

T. D. Noe, Table of n, a(n) for n = 1..10000 (with help from the Sierpiński problem website)
Ray Ballinger and Wilfrid Keller, The Sierpiński Problem: Definition and Status
Seventeen or Bust, A Distributed Attack on the Sierpiński problem

For the corresponding primes see A050921.
Cf. A103964, A040081.
Cf. A033809, A046067 (odd n), A057192 (prime n).

Do[m = 0; While[ !PrimeQ[n*2^m + 1], m++ ]; Print[m], {n, 1, 110} ]
sm[n_]:=Module[{k=0},While[!PrimeQ[n 2^k+1],k++];k]; Array[sm,120] (* Harvey P. Dale, Feb 05 2020 *)

A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, 2, 1, 1, 4, 2, 5, 4, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 4, 2, 1, 8, 2, 1, 2, 1, 3, 16, 1, 3, 6, 1, 1, 2, 3, 1, 8, 6, 1, 2, 3, 1, 4, 1, 3, 2, 1, 53, 6, 8, 3, 4, 1, 1, 8, 6, 3, 2, 1, 7, 2, 8, 1, 2, 2, 1, 4, 1, 3, 6, 1, 1, 2, 4, 15, 2

Offset: 1

Eric W. Weisstein

sign

There exist odd integers 2k-1 such that (2k-1)2^n+1 is always composite.
The smallest known example is 78557. Therefore a(39279) = -1.
For the corresponding primes see A057025(n-1), n >= 1, where a 0 will show up if a(n) = -1. - Wolfdieter Lang, Feb 07 2013.
Jaeschke shows that every positive integer appears infinitely often. - Jeppe Stig Nielsen, Jul 06 2020

Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.

T. D. Noe, Table of n, a(n) for n = 1..5000 (with help from the Sierpiński problem website; typo in a(3707)=1 corrected by Jeppe Stig Nielsen)
Ray Ballinger and Wilfrid Keller, Sierpiński Problem
John R. Cowles and Ruben Gamboa, Verifying Sierpiński and Riesel Numbers in ACL2, arXiv preprint arXiv:1110.4671 [cs.DM], 2011.
G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.
Seventeen or Bust, A Distributed Attack on the Sierpiński Problem
W. Sierpiński, Sur un problème concernant les nombres k*2^n+1, Elem. d. Math. 15, pp. 73-74, 1960.
Eric Weisstein's World of Mathematics, Riesel Number.
Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind.

Cf. A046068.
Bisection of A040076. Cf. A033809.
Cf. A057192, A057025.

max = 10000 (* this maximum value of m is sufficient up to n = 1000 *); a[n_] := For[m = 1, m <= max, m++, If[PrimeQ[(2n - 1)*2^m + 1], Return[m]]] /. Null -> -1; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 08 2012 *)

A250204 Sierpiński problem in base 6: Least k > 0 such that n*6^k+1 is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 4, 0, 1, 1, 1, 1, 0, 2, 1, 1, 1, 0, 5, 1, 4, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 1, 0, 1, 2, 1, 1, 0, 1, 5, 5, 2, 0, 1, 1, 1, 3, 0, 2, 1, 1, 7, 0, 1, 1, 2, 1, 0, 2, 1, 1, 1, 0, 2, 1, 8, 1, 0, 1, 2, 1, 1, 0, 7, 1, 1, 4, 0, 4, 1, 2, 1, 0, 2, 5, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 9, 2, 0, 1, 1, 1, 1, 0, 1, 6, 1, 2, 0, 1, 3, 1, 4, 0, 1, 2, 23, 1, 0, 4

Offset: 1

Eric Chen, Mar 11 2015

nonn

a(5k+4) = 0, since (5k+4)*6^n+1 is always divisible by 5, but there are infinitely many numbers not in the form 5k+4 such that a(n) = 0. For example, a(174308) = 0 since 174308*6^n+1 is always divisible by 7, 13, 31, 37, or 97 (See A123159). Conjecture: if n is not in the form 5k+4 and n < 174308, then a(n) > 0.

However, according to the Barnes link no primes n*6^k+1 are known for n = 1296, 7776 and 46656, so these may be counterexamples. - Robert Israel, Mar 17 2015

Eric Chen, Table of n, a(n) for n = 1..1000
Gary Barnes, Sierpinski conjectures and proofs

Cf. A040076, A046067, A078680, A033809, A123159.

Cf. A250205 (Least k > 0 such that n*6^k-1 is prime).

N:= 1000: # to get a(1) to a(N), using k up to 10000
a[1]:= 1:
for n from 2 to N do
  if n mod 5 = 4 then a[n]:= 0
  else
    for k from 1 to 10000 do
    if isprime(n*6^k+1) then
       a[n]:= k;
       break
    fi
    od
  fi
od:
L:= [seq(a[n],n=1..N)]; # Robert Israel, Mar 17 2015

(* m <= 10000 is sufficient up to n = 1000 *)
a[n_] := For[k = 1, k <= 10000, k++, If[PrimeQ[n*6^k + 1], Return[k]]] /. Null -> 0; Table[a[n], {n, 1, 120}]

a(n) = if(n%5==4, 0, for(k = 1, 10000, if(ispseudoprime(n*6^k+1), return(k))))

A215540 Least k such that (2n-1)2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

Original entry on oeis.org

1, 41, 7, 14, 67, 18759, 20, 229, 147, 6838, 41

Offset: 1

Arkadiusz Wesolowski, Aug 15 2012

(2*n-1)*2^a(n) + 1 is in A023394.
a(n) >= 7 for n > 1.
a(39279) = 0. No n < 39279 with a(n)=0 is known.
a(12)>2500000, a(13)>2500000, a(14)=455, a(15)=57 (see Ballinger and Keller link).
No, a(13)=2141884, found in 2011. - Jeppe Stig Nielsen, Sep 07 2019

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300
Fermat factoring status, Prime factors of Fermat numbers
FermatSearch, Home page
PrimeGrid, Announcement of 25*2^2141884+1, related to a(13).
Eric Weisstein's World of Mathematics, Fermat Number

Cf. A000215, A033809.

lst = {}; Do[k = 1; While[True, p = n*2^k + 1; If[PrimeQ[p] && IntegerQ@Log[2, MultiplicativeOrder[2, p]], AppendTo[lst, k]; Break[]]; k++], {n, 1, 9, 2}]; lst

A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2n-1)2^k+1, if they exist and n > 1; and of zeros otherwise.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 2, 3, 5, 8, 1, 4, 7, 6, 16, 1, 2, 6, 13, 8, 32, 2, 3, 3, 14, 15, 12, 64, 1, 8, 5, 6, 20, 25, 18, 128, 3, 2, 10, 7, 7, 26, 39, 30, 256, 6, 15, 4, 20, 19, 11, 50, 55, 36, 512, 1, 10, 27, 9, 28, 21, 14, 52, 75, 41, 1024, 1, 4, 46, 51, 10, 82, 43, 17, 92, 85, 66, 2048

Offset: 1

Lorenzo Sauras Altuzarra, Mar 01 2023

Is a(n) <= A279709(n)?

			Table starts
  1   2   4   8  16  32  64 128 ... A000079
  1   2   5   6   8  12  18  30 ... A002253
  1   3   7  13  15  25  39  55 ... A002254
  2   4   6  14  20  26  50  52 ... A032353
  1   2   3   6   7  11  14  17 ... A002256
  1   3   5   7  19  21  43  81 ... A002261
  2   8  10  20  28  82 188 308 ... A032356
  1   2   4   9  10  12  27  37 ... A002258
  ...
(2*39279 - 1)*2^r + 1 is composite for every r > 0 (see comments from A046067), so the 39279th row is A000004, the zero sequence.

Ray Ballinger and Wilfrid Keller, List of primes k.2^n + 1 for k < 300.

Cf. A000079, A002253, A002254, A032353, A002256, A002261, A032356, A002258.
Cf. A033809 (1st column).
Cf. A000004, A046067, A279709.

vk(k, nn) = if (k==1, return (vector(nn, i, 2^(i-1)))); my(v = vector(nn-k+1), nb=0, i=0, x); while (nb != nn-k+1, if (isprime((2*k-1)*2^i+1), nb++; v[nb] = i); i++;); v;
lista(nn) = my(v=vector(nn, k, vk(k, nn))); my(w=List()); for (i=1, nn, for (j=1, i, listput(w, v[i-j+1][j]););); Vec(w); \\ Michel Marcus, Mar 03 2023

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A029810 Erroneous version of A033809, A046067.

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A040076 Smallest m >= 0 such that n*2^m + 1 is prime, or -1 if no such m exists.

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A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists.

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A250204 Sierpiński problem in base 6: Least k > 0 such that n*6^k+1 is prime, or 0 if no such k exists.

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A215540 Least k such that (2*n-1)*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

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A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2*n-1)*2^k+1, if they exist and n > 1; and of zeros otherwise.

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PARI

A215540 Least k such that (2n-1)2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m, or 0 if no such value exists.

A361076 Array, read by ascending antidiagonals, whose n-th row consists of the powers of 2, if n = 1; of the primes of the form (2n-1)2^k+1, if they exist and n > 1; and of zeros otherwise.