A205322 -id:A205322 - cp's OEIS frontend

A205321 Smallest k>=0 such that (2^n+k)2^n-1 and (2^n+k)2^n+1 are a twin prime pair.

0, 11, 1, 11, 4, 11, 13, 116, 34, 14, 241, 44, 97, 458, 337, 59, 604, 206, 67, 167, 424, 179, 97, 326, 259, 284, 1177, 77, 328, 356, 508, 74, 1798, 749, 2197, 1289, 643, 839, 1171, 1427, 814, 2564, 31, 4244, 379, 5099, 3706, 4871, 2719, 3194, 7057, 122, 5329, 2636, 301, 2852, 3793

Offset: 1

Pierre CAMI, Jul 14 2012

nonn

Conjecture : there is at least one k for each n.

Pierre CAMI, Table of n, a(n) for n = 1..825

Cf. A191617, A191618, A205322

A205321 := proc(n)
    local a,p ;
    for a from 0 do
         p := (2^n+a)*2^n-1 ;
        if isprime(p) and isprime(p+2) then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 18 2012

a(n) = A082466(2^n), n>1. - R. J. Mathar, Jul 20 2012

A212038 The size of the set of numbers k>=0 such that all (2^n-k)2^n-1 are prime but only (2^n-k)2^n+1 with the largest k is also the associated twin prime.

Original entry on oeis.org

1, 7, 5, 2, 4, 11, 6, 1, 3, 2, 31, 2, 11, 11, 11, 6, 30, 16, 2, 36, 90, 56, 11, 52, 13, 36, 10, 62, 20, 31, 23, 28, 30, 14, 47, 22, 10, 5, 104, 39, 11, 14, 64, 184, 209, 176, 193, 162, 25, 38, 23, 5, 27, 157, 5, 17, 32, 90, 1, 199, 96, 83, 29, 82, 12, 220, 19, 40, 37, 13, 16, 120, 11, 130, 12, 77, 202

Offset: 4

Pierre CAMI, Jul 14 2012

nonn

Search set similar to A212037 but the sign of k in the prime form is switched.

Heuristically the average of a(n)/n for n=1 to N tends to 1 as N increases.

Pierre CAMI, Table of n, a(n) for n = 4..825

Cf. A191619, A191620, A205322.

A212038 := proc(n)
    local a,k,p ;
    a := 0 ;
    for k from 0 do
        p := (2^n-k)*2^n-1 ;
        if isprime(p) then
            a := a+1 ;
        end if;
        if isprime(p) and isprime(p+2) then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 21 2012

SCRIPT
DIM nn, 3
DIM jj
DIM kk
DIMS tt
OPENFILEOUT myfile, a(n).txt
LABEL loopn
SET nn, nn+1
IF nn>825 THEN END
SET kk, -1
SET jj, 0
LABEL loopk
SET kk, kk+1
SETS tt, %d, %d\,; nn; kk
PRP (2^nn-kk)*2^nn-1, tt
IF ISPRP THEN GOTO a
IF ISPRIME THEN GOTO a
GOTO loopk
LABEL a
SET jj, jj+1
PRP (2^nn-kk)*2^nn+1, tt
IF ISPRP THEN GOTO d
IF ISPRIME THEN GOTO d
GOTO loopk
LABEL d
SETS tt, %d, %d\,; nn; jj
WRITE myfile, tt
GOTO loopn

A216217 Smallest k such that 6^n - 2k3^n - 1 and 6^n - 2k3^n + 1 are twin primes or 0 if no solution, n > 1.

Original entry on oeis.org

1, 2, 3, 0, 3, 11, 33, 9, 26, 6, 34, 138, 51, 19, 33, 246, 66, 31, 167, 73, 13, 716, 138, 148, 138, 339, 447, 41, 131, 41, 9, 178, 778, 337, 543, 2154, 213, 1216, 454, 183, 678, 442, 157, 381, 297, 1476, 54, 1201, 1942, 1566, 572, 3708, 3261, 3672, 1087, 306

Offset: 2

Pierre CAMI, Mar 13 2013

nonn

Conjecture: there is only one zero term: a(5) = 0.

The PFGW script computes 2*a(n).

			6^2 - 2*1*3^2 - 1 = 17, 17 and 19 twin primes so a(2)=1.
6^3 - 2*2*3^3 - 1 = 107, 107 and 109 twin primes so a(3)=2.
6^4 - 2*3*3^4 - 1 = 809, 809 and 811 twin primes so a(4)=3.
6^5 - 2*k*3^5 - 1 and 6^5 - 2*k*3^5 + 1 for k=1 to 30 have no twin prime solution so a(5)=0.

Pierre CAMI, Table of n, a(n) for n = 2..400

Cf. A205322 (similar, but powers of 2).

Table[k = 0; While[k++; p = 6^n - 2*k*3^n - 1; p > 0 && ! (PrimeQ[p] && PrimeQ[p + 2])]; If[p <= 0, 0, k], {n, 2, 50}] (* T. D. Noe, Mar 15 2013 *)

cp's OEIS Frontend

A205321 Smallest k>=0 such that (2^n+k)2^n-1 and (2^n+k)2^n+1 are a twin prime pair.

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A212038 The size of the set of numbers k>=0 such that all (2^n-k)2^n-1 are prime but only (2^n-k)2^n+1 with the largest k is also the associated twin prime.

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Maple

PFGW

A216217 Smallest k such that 6^n - 2k3^n - 1 and 6^n - 2k3^n + 1 are twin primes or 0 if no solution, n > 1.

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Author

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cp's OEIS Frontend

A205321 Smallest k>=0 such that (2^n+k)*2^n-1 and (2^n+k)*2^n+1 are a twin prime pair.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A212038 The size of the set of numbers k>=0 such that all (2^n-k)*2^n-1 are prime but only (2^n-k)*2^n+1 with the largest k is also the associated twin prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PFGW

A216217 Smallest k such that 6^n - 2*k*3^n - 1 and 6^n - 2*k*3^n + 1 are twin primes or 0 if no solution, n > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A205321 Smallest k>=0 such that (2^n+k)2^n-1 and (2^n+k)2^n+1 are a twin prime pair.

A212038 The size of the set of numbers k>=0 such that all (2^n-k)2^n-1 are prime but only (2^n-k)2^n+1 with the largest k is also the associated twin prime.

A216217 Smallest k such that 6^n - 2k3^n - 1 and 6^n - 2k3^n + 1 are twin primes or 0 if no solution, n > 1.