A205321 -id:A205321 - cp's OEIS frontend

A212037 The size of the set of numbers k>=0 such that all (2^n+k)2^n-1 are prime but only the last (largest) (2^n+k)2^n+1 is also an associated twin prime.

1, 6, 1, 4, 2, 2, 2, 24, 6, 2, 28, 7, 16, 47, 29, 6, 41, 16, 3, 17, 32, 10, 10, 23, 14, 15, 52, 4, 13, 20, 23, 4, 84, 26, 88, 50, 20, 35, 51, 44, 41, 87, 1, 142, 13, 188, 107, 162, 91, 96, 197, 4, 148, 71, 9, 66, 97, 41, 10, 9, 152, 234, 48, 104, 144, 40, 18, 45, 52, 204, 21, 49, 51, 9, 102, 13, 31, 108, 88

Offset: 1

Pierre CAMI, Jul 14 2012

nonn

Starting at a count of zero, we consider for increasing k>=0 the pairs (2^n+k)*2^n+-1. If the smaller of these two numbers is prime, we increase the counter. If the larger of these two numbers is also prime, we admit the counter to the sequence. It is basically a measure of how many unsuccessful primality tests on the larger of the two numbers are done before it becomes a compatible twin prime.

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

			For n=2, the 6 pairs (19,21) at k=1, (23,25) at k=2, (31,33) at k=4, (43,45) at k=7, (47,49) at k=8 and (59,61) at k=11 are counted. The smaller of these must be a prime to be counted, and at k=11 also the larger (i.e., 61) becomes prime, which finishes the search.

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

Cf. A191617, A191218, A205321.

A212037 := 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 20 2012

A205322 Smallest k>=0 such that (2^n-k)2^n-1 and (2^n-k)2^n+1 are a twin prime pair; or -1 if no such k exists.

Original entry on oeis.org

0, 1, -1, 1, 26, 22, 8, 28, 47, 16, 14, 19, 17, 316, 8, 133, 116, 166, 77, 364, 197, 49, 647, 1594, 848, 109, 869, 169, 773, 166, 1274, 466, 512, 694, 644, 733, 401, 1636, 662, 184, 71, 3106, 1157, 346, 332, 2194, 6179, 7999, 6023, 6784, 5612, 1108, 1001, 649, 197

Offset: 1

Pierre CAMI, Jul 14 2012

sign

Conjecture: there is always at least one k>=0 unless n=3.

Pierre CAMI, Table of n, a(n) for n = 1..826

Cf. A191619, A191620, A205321.

A205322 := proc(n)
    local a,p ;
    for a from 0 to 2^n do
         p := (2^n-a)*2^n-1 ;
        if isprime(p) and isprime(p+2) then
            return a;
        end if;
    end do:
    return -1 ;
end proc: # R. J. Mathar, Jul 18 2012

Table[k = -1; While[k++; p = 4^n - k*2^n - 1; p > 0 && ! (PrimeQ[p] && PrimeQ[p + 2])]; If[p <= 0, -1, k], {n, 50}] (* T. D. Noe, Mar 15 2013 *)

cp's OEIS Frontend

A212037 The size of the set of numbers k>=0 such that all (2^n+k)2^n-1 are prime but only the last (largest) (2^n+k)2^n+1 is also an associated twin prime.

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A205322 Smallest k>=0 such that (2^n-k)2^n-1 and (2^n-k)2^n+1 are a twin prime pair; or -1 if no such k exists.

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cp's OEIS Frontend

A212037 The size of the set of numbers k>=0 such that all (2^n+k)*2^n-1 are prime but only the last (largest) (2^n+k)*2^n+1 is also an associated twin prime.

Views

Author

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Comments

Examples

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Crossrefs

Programs

Maple

A205322 Smallest k>=0 such that (2^n-k)*2^n-1 and (2^n-k)*2^n+1 are a twin prime pair; or -1 if no such k exists.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Maple

Mathematica

A212037 The size of the set of numbers k>=0 such that all (2^n+k)2^n-1 are prime but only the last (largest) (2^n+k)2^n+1 is also an associated twin prime.

A205322 Smallest k>=0 such that (2^n-k)2^n-1 and (2^n-k)2^n+1 are a twin prime pair; or -1 if no such k exists.