A046067 -id:A046067 - cp's OEIS frontend

A238904 Smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k exists.

0, 1, 1, 2, 1, 1, 2, 1, -1, 6, 1, -1, 2, 2, 1, -1, 6, 1, 2, 1, 1, 2, 9, -1, 2, 1, -1, 4, 2, -1, 12, 4, 1, 2, 1, 3, 6, 3, -1, 2, 1, -1, 4, 6, 9, 8, 2, 1, 2, 1, 3, -1, 1, -1, 6, 1, -1, 12, 6, 3, 12, 8, 1, 2, 3, 3, 4, 1, -1, -1, 3, -1, -1, 60, 3, 4, 2, 1, 12

Offset: 0

Ilya Lopatin and Juri-Stepan Gerasimov, Mar 07 2014

sign

			a(0) = 0 because 2^0 + (2*0+1) = 2 and (2*0+1)*2^0 + 1 = 2 are both prime.

Giovanni Resta, Table of n, a(n) for n = 0..1000

Cf. A046067, A238797.

a[n_] := Catch@Block[{k=0}, While[k <= n, If[PrimeQ[2^k + 2*n + 1] && PrimeQ[(2*n + 1)*2^k + 1], Throw@k]; k++]; -1]; a/@ Range[0,80] (* Giovanni Resta, Mar 15 2014 *)

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))))

A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 3, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 1, 2, 4, 1, 2, 4, 1, 1, 2, 3, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 4, 1, 2, 4, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 4, 3, 4, 4, 47, 1, 2, 1, 2, 6, 1, 1, 2, 3, 3, 8, 1, 1, 2, 3, 1, 2, 5, 1, 2, 1, 2, 4

Offset: 1

Eric Chen, Dec 14 2014

It is known that a(254602) = -1, because |509203-2^k| is always divisible by 3, 5, 7, 13, 17, or 241. a(1147) is the first unknown term.
a((A101036(n)+1)/2) = -1, so there are infinitely many n such that a(n) = -1.
a((A133122(n)+1)/2) = A096502((A133122(n)-1)/2).

			a(12) = 2 because 2*12-1 = 23 and that 23-2^1 = 21 is not prime but 23-2^2 = 19 is.
a(69) = 6 because 2*69-1 = 137, |137-2^k| is composite for k = 1, 2, 3, 4, 5 and prime for k = 6.
Even the smallest k can be also very large. For example, a(169) = 791.
a(1147) > 65536.

Jinyuan Wang, Table of n, a(n) for n = 1..1146

Cf. A006285, A096502, A133122, A188903.

Cf. A046067, A046069, A067760.

Table[k = 1; While[!PrimeQ[Abs[(2*n-1) - 2^k]], k++]; k, {n, 1, 1000}]

A252168(n)={ my(k=1); n=2*n-1; while(!ispseudoprime(abs(n-2^k)), k++); k }

a(19) corrected by Jinyuan Wang, Mar 25 2023

A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 if no such value exists.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 4, 0, 0, 2, 0, 2, 1, 0, 1, 9, 0, 0, 4, 1, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 4, 0, 1, 1, 1, 0, 2, 0, 0, 1, 0, 0, 1, 0, 3, 1, 2, 4, 1, 1, 0, 2, 0, 1, 5, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 80, 0, 6, 0, 8, 2, 0, 1

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 primes see A291438.
a(A005097(n)) = 0 and a(A047845(n+1)) > 0 (or = -1).

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

Cf. A005097, A046067, A047845, A123159, A291438.

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),t]:
od:
a;

Table[SelectFirst[Range[0, 10^3], PrimeQ[2 n*3^# + 1] &] /. k_ /; MissingQ@ k -> -1, {n, 104}] (* Michael De Vlieger, Aug 23 2017 *)

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

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

cp's OEIS Frontend

A238904 Smallest k such that 2^k + (2n+1) and (2n+1)*2^k + 1 are both prime, k <= n, or -1 if no such k 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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A252168 Smallest k > 0 such that |(2n-1) - 2^k| is prime, or -1 if no such k exists.

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A291437 Smallest m >= 0 such that (2*n)*3^m + 1 is prime, or -1 if no such value exists.

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Maple

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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

A291437 Smallest m >= 0 such that (2n)3^m + 1 is prime, or -1 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.