A264097 -id:A264097 - cp's OEIS frontend

A282428 Numbers k such that A264097(k) = A264098(k), so : A264097(k)2^k-1 and A264098(k)2^k+1 are twin primes.

1, 2, 6, 10, 18, 29, 63, 155, 211, 264, 546, 1032, 1156, 1321, 1553, 3460, 4901, 5907, 8335, 8529, 11455, 13153

Offset: 1

Pierre CAMI, Feb 15 2017

a(23) is > 31000.

The A264097(k) are : 3, 3, 3, 15, 3, 45, 9, 105, 9, 165, 297, 177, 1035, 1065, 291, 2403, 2565, 5775, 3975, 459, 915, 3981

Cf. A264097, A264098.

With[{nn = 600}, Flatten@ Position[Transpose@ {Table[k = 3; While[! PrimeQ[k 2^n - 1], k += 6]; k, {n, nn}], Table[k = 3; While[! PrimeQ[k 2^n + 1], k += 6]; k, {n, nn}]}, w_ /; SameQ @@ w]] (* Michael De Vlieger, Feb 16 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

cp's OEIS Frontend

A282428 Numbers k such that A264097(k) = A264098(k), so : A264097(k)2^k-1 and A264098(k)2^k+1 are twin primes.

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

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cp's OEIS Frontend

A282428 Numbers k such that A264097(k) = A264098(k), so : A264097(k)*2^k-1 and A264098(k)*2^k+1 are twin primes.

Views

Author

Keywords

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Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A282428 Numbers k such that A264097(k) = A264098(k), so : A264097(k)2^k-1 and A264098(k)2^k+1 are twin primes.