A057778 -id:A057778 - cp's OEIS frontend

A295640 Smallest k not divisible by 5 such that k*5^n + 1 is prime.

2, 4, 2, 18, 12, 4, 12, 24, 26, 18, 42, 28, 2, 16, 18, 34, 92, 4, 12, 46, 26, 16, 6, 36, 26, 16, 6, 16, 152, 18, 42, 136, 6, 162, 132, 58, 24, 142, 48, 22, 56, 16, 36, 84, 2, 12, 24, 108, 168, 4, 12, 214, 36, 114, 12, 444, 26, 54, 416, 42

Offset: 1

Pierre CAMI, Nov 25 2017

nonn

The ratio (Sum_(n=1..t) a(n)) / (Sum_(n=1..t) n) tends to log(5) as t increases.

Pierre CAMI, Table of n, a(n) for n = 1..2999

Cf. A057778, A295639, A295641.

Array[Block[{k = 2}, While[Or[Divisible[k, 5], ! PrimeQ[k 5^# + 1]], k++]; k] &, 60] (* Michael De Vlieger, Dec 18 2017 *)

a(n) = {k = 1; while (!isprime(k*5^n+1), k++; if (!(k%5), k++)); k;} \\ Michel Marcus, Nov 25 2017

A295641 Smallest k not divisible by 7 such that k*7^n + 1 is prime.

Original entry on oeis.org

4, 4, 4, 6, 10, 4, 4, 36, 6, 4, 4, 82, 36, 10, 24, 90, 4, 48, 12, 16, 118, 10, 72, 16, 94, 18, 24, 150, 76, 58, 34, 40, 156, 34, 52, 166, 16, 4, 36, 90, 120, 78, 34, 36, 10, 244, 60, 102, 18, 114, 172, 48, 94, 10, 66, 396, 234, 240, 166

Offset: 1

Pierre CAMI, Nov 25 2017

nonn

The ratio (Sum_(n=1..t) a(n)) / (Sum_(n=1..t) n) tends to log(7) as t increases.

Pierre CAMI, Table of n, a(n) for n = 1..2000

Cf. A057778, A295639, A295640.

Array[Block[{k = 2}, While[Or[Divisible[k, 7], ! PrimeQ[k 7^# + 1]], k++]; k] &, 59] (* Michael De Vlieger, Dec 18 2017 *)

a(n) = {k = 1; while (!isprime(k*7^n+1), k++; if (! (k%7), k++)); k;} \\ Michel Marcus, Nov 25 2017

A264098 Smallest odd number k divisible by 3 such that k*2^n + 1 is prime.

Original entry on oeis.org

3, 3, 9, 15, 3, 3, 9, 3, 15, 15, 9, 3, 33, 9, 81, 21, 9, 3, 27, 27, 33, 27, 45, 45, 33, 27, 15, 33, 45, 3, 39, 81, 9, 75, 81, 3, 15, 15, 81, 27, 3, 9, 9, 15, 189, 27, 27, 15, 105, 27, 75, 93, 51, 177, 57, 27, 75, 99, 27, 45, 105, 105, 9, 27, 9, 3, 9, 237

Offset: 1

Pierre CAMI, Nov 03 2015

nonn

As N increases, (Sum_{n=1..N} a(n))/(Sum_{n=1..N} n) appears to approach 2*log(2), as can be seen by plotting the first 31000 terms.

This observation is consistent with the prime number theorem as the probability that k*2^n+1 is prime is 1/(n*log(2)+log(k))/2 for k multiple of 3 so ~ 1/(2*n*log(2)) as n increases, if k ~ 2*n*log(2) then k/(2*n*log(2)) ~ 1.

			3*2^1 + 1 = 7 is prime so a(1) = 3.
3*2^2 + 1 = 13 is prime so a(2) = 3.
3*2^3 + 1 = 25 is composite; 9*2^3 + 1 = 73 is prime so a(3) = 9.

Pierre CAMI, Table of n, a(n) for n = 1..31000

Cf. A035050, A057778, A264097.

for n from 1 to 100 do
  for k from 3 by 6 do
    if isprime(k*2^n+1) then
      A[n]:= k; break
   fi
 od
od:
seq(A[n],n=1..100); # Robert Israel, Jan 22 2016

Table[k = 3; While[! PrimeQ[k 2^n + 1], k += 6]; k, {n, 68}] (* Michael De Vlieger, Nov 03 2015 *)

a(n) = {k = 3; while (!isprime(k*2^n+1), k += 6); k;} \\ Michel Marcus, Nov 03 2015

A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

Original entry on oeis.org

67, 9431, 461081, 2543551

Offset: 1

Arkadiusz Wesolowski, Feb 21 2017

Fernando (Remark 5.2) shows that all terms are odd. - Jeppe Stig Nielsen, Jan 02 2025

R. Fernando, Morehead-like restrictions on Fermat divisors.
Proth Search Page, Prime factors of Fermat numbers
Eric Weisstein's World of Mathematics, Fermat Number

Subsequence of A002256.

Cf. A000215, A050528, A057778, A201364, A204620, A226366, A280003.

A334296 Smallest k such that (2k+1)*2^n+1 is prime.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 1, 2, 0, 7, 6, 4, 1, 2, 3, 2, 0, 4, 1, 5, 3, 5, 12, 22, 22, 2, 3, 7, 6, 11, 1, 17, 21, 4, 37, 29, 1, 7, 7, 2, 13, 1, 4, 4, 7, 17, 9, 13, 7, 11, 3, 8, 3, 25, 24, 2, 13, 14, 49, 13, 15, 26, 52, 4, 12, 4, 1, 4, 15, 11, 19, 19, 63, 11, 33, 2, 46

Offset: 0

Mike Speciner, Apr 21 2020

nonn

A057775 is the corresponding sequence of primes.

			a(0)=a(1)=a(2)=0 because 2^0+1=2, 2^1+1=3, 2^2+1=5 are prime.
a(3)=2 because 2^8+1=9 and 3*2^8+1=25 are not prime, but 5*2^8+1=41 is.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A057778, A057775, A035050, A035089.

f:= proc(n) local t, v, k;
   t:= 2^n; v:= -t+1;
   for k from 0 do
      v:= v+2*t;
      if isprime(v) then return k fi
   od
end proc:
map(f, [$0..100]); # Robert Israel, Jul 14 2020

a[n_] := Block[{k = 0}, While[! PrimeQ[(2 k + 1) 2^n + 1], k++]; k]; Array[a, 77, 0] (* Giovanni Resta, May 08 2020 *)

a(n) = my(k=0); while (!isprime((2*k+1)*2^n+1), k++); k; \\ Michel Marcus, Apr 30 2020

from itertools import count
from sympy import isprime
def pow2p1() : # generates the sequence
  for n in count() :
    for k in count() :
      if isprime(((2*k+1)<

a(n) = (A057778(n)-1)/2.

a(n) = ((A057775(n)-1)/2^n-1)/2.

cp's OEIS Frontend

A295640 Smallest k not divisible by 5 such that k*5^n + 1 is prime.

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A295641 Smallest k not divisible by 7 such that k*7^n + 1 is prime.

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A264098 Smallest odd number k divisible by 3 such that k*2^n + 1 is prime.

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A280004 Numbers k such that 9*2^k + 1 is a prime factor of a Fermat number 2^(2^m) + 1 for some m.

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A334296 Smallest k such that (2k+1)*2^n+1 is prime.

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