A057025 -id:A057025 - cp's OEIS frontend

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

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 *)

A057026 Smallest prime of form (2n+1)*2^m-1 for some m, or 0 if no such prime exists.

Original entry on oeis.org

3, 2, 19, 13, 17, 43, 103, 29, 67, 37, 41, 367, 199, 53, 463, 61, 131, 139, 73, 311, 163, 5503, 89, 751, 97, 101, 211, 109, 113, 241663, 487, 251, 1039, 2143, 137, 283, 9343, 149, 307, 157, 647, 331, 2719, 173, 1423, 181, 743, 379, 193, 197, 103423, 823, 419

Offset: 0

Henry Bottomley, Jul 24 2000

nonn

If a(329) > 0 it is greater than 659*2^10000. - Robert Israel, Jul 01 2014
Indeed, a(329) > 659*2^100000 if it is nonzero. There does not appear to be a covering set, though, so probably a(329) > 0. - Charles R Greathouse IV, Jul 02 2014
a(329) = 659*2^800516 - 1 (found by David W Linton in 2004). - Robert Israel, Jul 04 2014

			a(5)=43 because 2*5+1=11 and smallest prime of the form 11*2^m-1 is 43 (since 10 and 21 are not prime)

Cf. A057024, A057025, A038699.

A057026:= proc(n)
local t;
     t:= 2*n;
     while not isprime(t) do t:= 2*t+1 od;
     t
end proc;
seq(A057026(n),n=0..328); # Robert Israel, Jul 01 2014

A291438 Smallest prime of the form (2n)3^m + 1 for some m >= 0, or -1 if no such prime exists.

Original entry on oeis.org

3, 5, 7, 73, 11, 13, 43, 17, 19, 61, 23, 73, 79, 29, 31, 97, 103, 37, 3079, 41, 43, 397, 47, 433, 151, 53, 163, 1102249, 59, 61, 5023, 193, 67, 613, 71, 73, 223, 229, 79, 241, 83, 757, 6967, 89, 271, 277, 283, 97, 883, 101, 103, 313, 107, 109, 331, 113, 3079

Offset: 1

Martin Renner, Aug 23 2017

nonn

There exist even integers 2*n such that (2*n)*3^m + 1 is always composite.
It is conjectured that the smallest one is 125050976086 = A123159(3), therefore a(62525488043) = -1.
For the corresponding numbers m see A291437.

			a(4) = 73 because 8*3^2 + 1 = 73 is the smallest prime of this form, since 8*3^0 + 1 = 9 and 8*3^1 + 1 = 25 are not prime.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A057025, A123159, A291437.

a:=[]:
for n from 1 to 10^3 do
  t:=-1:
  for m from 0 to 10^3 do # this max value of m is sufficient up to n=10^3
    if isprime((2*n)*3^m+1) then t:=m: break: fi:
  od:
  a:=[op(a),(2*n)*3^t+1]:
od:
a;

Table[If[# < 0, #, 1 + 2 n*3^#] &@ SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 60}] (* Michael De Vlieger, Aug 23 2017 *)

a(n) = {my(m = 0); while (!isprime(p=(2*n)*3^m + 1), m++); p;} \\ Michel Marcus, Aug 25 2017

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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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A057026 Smallest prime of form (2n+1)*2^m-1 for some m, or 0 if no such prime exists.

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A291438 Smallest prime of the form (2n)3^m + 1 for some m >= 0, or -1 if no such prime exists.

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cp's OEIS Frontend

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

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Mathematica

A057026 Smallest prime of form (2n+1)*2^m-1 for some m, or 0 if no such prime exists.

Views

Author

Keywords

Comments

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Crossrefs

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Maple

A291438 Smallest prime of the form (2*n)*3^m + 1 for some m >= 0, or -1 if no such prime exists.

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Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

PARI

A291438 Smallest prime of the form (2n)3^m + 1 for some m >= 0, or -1 if no such prime exists.