A214495 -id:A214495 - cp's OEIS frontend

A214496 Smallest k>0 such that (3^n+k)3^n-1 and (3^n+k)3^n+1 are a twin prime pair.

1, 3, 3, 7, 21, 29, 113, 31, 61, 13, 179, 237, 33, 201, 613, 171, 347, 291, 907, 437, 523, 193, 1039, 729, 567, 231, 1847, 931, 1023, 821, 329, 3937, 6137, 319, 1663, 667, 1837, 529, 1769, 1959, 1781, 743, 3223, 591, 613, 473, 5679, 2137, 567, 459, 4729

Offset: 1

Pierre CAMI, Jul 20 2012

nonn

Conjecture : there is always one such k for each n>0.

Heuristically, as N increases, the average of a(n)/n^2 for n=1 to N tends to 1.2

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A214495.

A214496 := proc(n)
    local k;
    for k from 1 do
        p := (3^n+k)*3^n-1 ;
        if isprime(p) and isprime(p+2) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 23 2012

A214497 Smallest k>=0 such that (3^n-k)2^n-1 and (3^n-k)2^n+1 are a twin prime pair.

Original entry on oeis.org

0, 6, 3, 9, 9, 6, 3, 93, 3, 54, 18, 96, 213, 297, 1206, 258, 312, 201, 261, 1206, 1158, 396, 1062, 216, 708, 762, 816, 678, 3579, 762, 831, 2106, 4734, 576, 333, 633, 213, 2766, 363, 2454, 1464, 2007, 4551, 3183, 1497, 4899, 198, 66, 9984, 2847, 276, 3051

Offset: 1

Pierre CAMI, Jul 20 2012

nonn

Conjecture : there is always one such k for each n>0.

As N increases, the average of a(n)/n^2 over n=1 to N appears to approach 1.1

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A214495-A214498.

A214497 := proc(n)
    local k;
    for k from 0 do
        p := (3^n-k)*2^n-1 ;
        if isprime(p) and isprime(p+2) then
            return k;
        end if;
    end do:
end proc:
seq(A214497(n),n=1..80) ; # R. J. Mathar, Jul 23 2012

sk[n_]:=Module[{k=0,c},c=(3^n-k)2^n;While[!PrimeQ[c-1] || !PrimeQ[c+1],k++;c=(3^n-k)2^n];k]; Array[sk,60] (* Harvey P. Dale, Dec 09 2012 *)

A214498 Smallest k>=0 such that (3^n+k)2^n-1 and (3^n+k)2^n+1 are a twin prime pair.

Original entry on oeis.org

0, 6, 3, 12, 78, 18, 18, 141, 18, 54, 78, 132, 138, 78, 57, 537, 237, 6, 972, 219, 81, 3369, 69, 501, 2328, 18, 738, 291, 393, 969, 324, 492, 102, 3291, 1788, 1401, 891, 954, 4017, 309, 702, 1656, 3999, 1014, 2346, 4008, 3, 5001, 2736, 558, 2262, 969, 762

Offset: 1

Pierre CAMI, Jul 20 2012

nonn

Conjecture : there is always one such k for each n>0.

Heuristically, the average of a(n)/n^2 over n=1 to N tends to 1.2 as N increases.

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A214495-A214497.

A214498 := proc(n)
    local k;
    for k from 0 do
        p := (3^n+k)*2^n-1 ;
        if isprime(p) and isprime(p+2) then
            return k;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 23 2012

sk[n_]:=Module[{n3=3^n,n2=2^n,k=0},While[!And@@PrimeQ[(n3+k)n2+{1,-1}], k++];k]; Array[sk,60] (* Harvey P. Dale, Jul 23 2013 *)

A214499 The size of the set of numbers k>=0 such that all (3^n-k)3^n-1 are prime but only the last (largest) (3^n-k)3^n+1 is also an associated twin prime.

Original entry on oeis.org

1, 1, 4, 4, 1, 1, 2, 3, 4, 6, 4, 6, 4, 11, 3, 10, 6, 9, 4, 18, 30, 16, 8, 29, 5, 32, 21, 15, 45, 5, 97, 10, 36, 3, 33, 35, 55, 20, 54, 25, 30, 30, 36, 8, 38, 30, 16, 6, 3, 20, 10, 35, 36, 2, 84, 20, 52, 85, 25, 25, 70, 46, 15, 53, 6, 103, 11, 27, 87, 15, 42, 14

Offset: 1

Pierre CAMI, Jul 20 2012

nonn

Starting at a count of zero, we consider for increasing k>=0 the pairs (3^n-k)*3^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 0.83 as N increases.

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A212037, A214495, A214496, A214500, A214501, A214502.

A214500 The size of the set of numbers k>=0 such that all (3^n+k)3^n-1 are prime but only the last (largest) (3^n+k)3^n+1 is also an associated twin prime.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 9, 3, 4, 1, 8, 20, 4, 10, 24, 8, 17, 9, 28, 16, 19, 6, 33, 19, 12, 5, 49, 27, 25, 26, 10, 91, 143, 9, 41, 14, 36, 11, 34, 26, 28, 10, 50, 7, 12, 11, 8, 27, 13, 4, 44, 138, 50, 10, 45, 21, 51, 84, 65, 48, 39, 139, 36, 19, 22, 85, 113, 105, 5, 36

Offset: 1

Pierre CAMI, Jul 20 2012

nonn

Starting at a count of zero, we consider for increasing k>=0 the pairs (3^n+k)*3^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 0.83 as N increases.

Cf. A212037, A214495, A214496, A214499.

cp's OEIS Frontend

A214496 Smallest k>0 such that (3^n+k)*3^n-1 and (3^n+k)*3^n+1 are a twin prime pair.

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A214497 Smallest k>=0 such that (3^n-k)*2^n-1 and (3^n-k)*2^n+1 are a twin prime pair.

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A214498 Smallest k>=0 such that (3^n+k)*2^n-1 and (3^n+k)*2^n+1 are a twin prime pair.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A214499 The size of the set of numbers k>=0 such that all (3^n-k)*3^n-1 are prime but only the last (largest) (3^n-k)*3^n+1 is also an associated twin prime.

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Author

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Comments

Links

Crossrefs

A214500 The size of the set of numbers k>=0 such that all (3^n+k)*3^n-1 are prime but only the last (largest) (3^n+k)*3^n+1 is also an associated twin prime.

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Author

Keywords

Comments

Crossrefs

A214496 Smallest k>0 such that (3^n+k)3^n-1 and (3^n+k)3^n+1 are a twin prime pair.

A214497 Smallest k>=0 such that (3^n-k)2^n-1 and (3^n-k)2^n+1 are a twin prime pair.

A214498 Smallest k>=0 such that (3^n+k)2^n-1 and (3^n+k)2^n+1 are a twin prime pair.

A214499 The size of the set of numbers k>=0 such that all (3^n-k)3^n-1 are prime but only the last (largest) (3^n-k)3^n+1 is also an associated twin prime.

A214500 The size of the set of numbers k>=0 such that all (3^n+k)3^n-1 are prime but only the last (largest) (3^n+k)3^n+1 is also an associated twin prime.