A247479 -id:A247479 - cp's OEIS frontend

A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

3, 3, 3, 3, 7, 3, 3, 5, 7, 5, 3, 5, 9, 5, 9, 17, 7, 3, 51, 17, 7, 33, 13, 39, 57, 11, 21, 27, 7, 213, 15, 5, 31, 3, 25, 17, 21, 3, 25, 107, 15, 33, 3, 35, 7, 23, 31, 5, 19, 11, 21, 65, 147, 5, 3, 33, 51, 77, 45, 17, 69, 53, 9, 3, 67, 63, 43, 63, 51, 27, 73, 5

Offset: 1

Pierre CAMI, Nov 25 2014

nonn

Limit_{N->oo} (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) = log(2). [[Is there a proof or is this a conjecture? - Peter Luschny, Feb 06 2015]]

Records: 3, 7, 9, 17, 51, 57, 213, 255, 267, 321, 615, 651, 867, 901, 909, 1001, 1255, 1729, 1905, 2163, 3003, 3007, 3515, 3797, 3825, 4261, 4335, 5425, 5717, 6233, 6525, 6763, 11413, 11919, 12935, 20475, 20869, 25845, 30695, 31039, 31309, 42991, 55999, ... . - Robert G. Wilson v, Feb 08 2015

Robert G. Wilson v, Table of n, a(n) for n = 1..10031 (first 5000 terms from Pierre CAMI)

Cf. A126715, A247479.

f:= proc(n)
local k,p;
  p:= 2^n;
for k from 3 by 2 do if isprime(k*p-1) then return k fi od;
end proc:
seq(f(n), n=1 .. 100); # Robert Israel, Feb 05 2015

f[n_] := Block[{k = 3, p = 2^n}, While[ !PrimeQ[k*p - 1], k += 2]; k]; Array[f, 70]

a(n) = {k=3; while (!isprime(k*2^n-1), k+=2); k;} \\ Michel Marcus, Nov 25 2014

a(A002235(n)) = 3.

A253398 Smallest odd k > 1 such that k*2^prime(n) + 1 is prime.

Original entry on oeis.org

3, 5, 3, 5, 9, 5, 9, 11, 45, 23, 35, 15, 3, 9, 27, 51, 27, 53, 9, 39, 23, 249, 51, 51, 131, 221, 29, 105, 321, 179, 5, 221, 111, 411, 191, 65, 83, 75, 95, 101, 147, 83, 149, 111, 203, 131, 9, 245, 281, 15, 83, 65, 299

Offset: 1

Pierre CAMI, Dec 31 2014

nonn

For n < 1275, the ratio a(n)/prime(n) is always < 6 and on average ~0.7.
From Robert G. Wilson v, Jan 27 2015: (Start)
Records: 3, 5, 9, 11, 45, 51, 53, 249, 321, 411, 611, 1383, 1875, 2423, 4239, 4623, 6549, 7095, 8091, 9003, 10065, 10719, 18005, 18545, 19251, 21111, 25409, 39741, 49709, 54455, ..., .
a(n)=3 for n = 1, 3, 13, 71, ...;
a(n)=5 for n = 2, 4, 6, 31, 466, ...;
a(n)=9 for n = 5, 7, 14, 19, 47, 342, 1167, ...;
a(n)=11 for n = 8, ...; etc.
(End)

			3*2^2 + 1 = 13 (prime), so a(1)=3.
3*2^3 + 1 = 25 (composite), 5*2^3 + 1 = 41 (prime), so a(2)=5.
3*2^5 + 1 = 97 (prime), so a(3)=3.

Robert G. Wilson v, Table of n, a(n) for n = 1..1500 (first 1275 terms from Pierre CAMI)

Cf. A247479, A253027.

f[n_] := Block[{k = 3, p = 2^Prime@ n}, While[ !PrimeQ[ k*p + 1], k += 2]; k]; Array[f, 53] (* Robert G. Wilson v, Jan 25 2015 *)

a(n)=k=1;while(!isprime((2*k+1)*2^prime(n)+1),k++);2*k+1
vector(100,n,a(n)) \\ Derek Orr, Dec 31 2014

a(n) = A247479(p_n). - Robert G. Wilson v, Jan 27 2015

A263046 Smallest number k>2 such that k*2^n + 1 is a prime number.

Original entry on oeis.org

4, 3, 3, 5, 6, 3, 3, 5, 3, 15, 12, 6, 3, 5, 4, 5, 12, 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

Offset: 0

Pierre CAMI, Oct 08 2015

nonn

If k = 2^j then 2^(n+j) + 1 is a Fermat prime.
a(n) = 3 if and only if 3*2^n + 1 is a prime; that is, n belongs to A002253. - Altug Alkan, Oct 08 2015
a(n+1) >= ceiling(a(n)/2). If a(n) is even then a(n+1) = a(n)/2. - Robert Israel, Oct 08 2015

			3*2^1 + 1 = 7 (prime), so a(1)=3:
3*2^2 + 1 = 13 (prime), so a(2)=3;
3*2^3 + 1 = 25 (composite), 4*2^3 + 1 = 33 (composite), 5*2^3 - 1 = 41 (prime), so a(3)=5.

Pierre CAMI, Table of n, a(n) for n = 0..10000

Cf. A247479, A262994.

f:= proc(n) local k;
    for k from 3 do if isprime(k*2^n+1) then return k fi od
  end proc:
seq(f(n),n=1..100); # Robert Israel, Oct 08 2015

Table[k = 3; While[! PrimeQ[k 2^n + 1], k++]; k, {n, 76}] (* Michael De Vlieger, Oct 08 2015 *)

a(n) = {k=3; while (! isprime(k*2^n+1), k++); k;} \\ Michel Marcus, Oct 08 2015

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A247202 Smallest odd k > 1 such that k*2^n - 1 is a prime number.

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A253398 Smallest odd k > 1 such that k*2^prime(n) + 1 is prime.

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A263046 Smallest number k>2 such that k*2^n + 1 is a prime number.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI