A191617 -id:A191617 - cp's OEIS frontend

A191620 Least k such that (2^n-k)*2^n - 1 is a prime number.

0, 1, 2, 1, 1, 2, 2, 7, 1, 1, 14, 2, 11, 11, 2, 22, 7, 1, 2, 8, 2, 11, 14, 32, 2, 13, 2, 11, 52, 8, 10, 49, 13, 11, 4, 11, 31, 1, 23, 64, 11, 47, 20, 38, 1, 14, 4, 88, 7, 1, 47, 14, 22, 8, 2, 14, 1, 31, 20, 71, 20, 4, 44, 101, 38, 43, 80, 49, 11, 59, 4, 8

Offset: 1

Pierre CAMI, Jun 09 2011

nonn

Does a(n) exist for every n? This does not seem to be known, even on the GRH; see Heath-Brown. [Charles R Greathouse IV, Dec 27 2011]

Pierre CAMI, Table of n, a(n) for n = 1..5000
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proceedings of the London Mathematical Society 64:3 (1992), pp. 265-338.

Cf. A191617, A191618, A191619, A191621.

Table[a = 0; While[! PrimeQ[(2^n - a)*2^n - 1], a++]; a, {n, 100}] (* T. D. Noe, Jun 11 2011 *)

a(n)=forstep(k=4^n-1,1,-2^n,if(ispseudoprime(k),return(2^n-(k+1)>>n))) \\ Charles R Greathouse IV, Dec 27 2011

A191618 Least a such that (2^n+a)*2^n - 1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 11, 2, 2, 1, 13, 2, 4, 22, 2, 1, 2, 2, 43, 4, 2, 13, 2, 1, 13, 1, 2, 46, 8, 29, 83, 2, 8, 34, 1, 11, 19, 31, 25, 7, 38, 31, 31, 76, 52, 31, 43, 32, 13, 92, 2, 1, 59, 22, 1, 16, 19, 11, 16, 74, 8, 13, 8, 74, 2, 121, 20, 49, 85, 134, 116, 16

Offset: 1

Pierre CAMI, Jun 09 2011

nonn

Pierre CAMI, Table of n, a(n) for n = 1..5000

Cf. A191617, A191619, A191620, A191621.

Table[a = 0; While[! PrimeQ[(2^n + a)*2^n - 1], a++]; a, {n, 100}] (* T. D. Noe, Jun 11 2011 *)

A191619 Least a such that (2^n-a)*2^n + 1 is a prime number.

Original entry on oeis.org

0, 0, 3, 0, 3, 10, 3, 0, 3, 10, 3, 4, 3, 4, 3, 16, 23, 4, 3, 21, 12, 10, 18, 40, 14, 37, 8, 16, 32, 10, 36, 1, 63, 10, 3, 48, 17, 67, 3, 31, 33, 22, 9, 19, 3, 9, 47, 33, 21, 15, 3, 58, 51, 22, 78, 163, 8, 30, 3, 85, 44, 4, 71, 28, 204, 4, 42, 75, 27, 16, 17

Offset: 1

Pierre CAMI, Jun 09 2011

nonn

Does a(n) exist for every n? This does not seem to be known, even on the GRH; see Heath-Brown. [Charles R Greathouse IV, Dec 27 2011]

Pierre CAMI, Table of n, a(n) for n = 1..5000
D. R. Heath-Brown, Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression. Proceedings of the London Mathematical Society 64:3 (1992), pp. 265-338.

Cf. A191617, A191618, A191620, A191621.

Table[a = 0; While[! PrimeQ[(2^n - a)*2^n + 1], a++]; a, {n, 100}] (* T. D. Noe, Jun 11 2011 *)

a(n)=forstep(k=4^n+1, 1, -2^n, if(ispseudoprime(k), return(2^n-(k-1)>>n))) \\ Charles R Greathouse IV, Dec 27 2011

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.

Original entry on oeis.org

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

A200920 Least k such that (3^n+ k)*3^n + 1 is a prime number.

Original entry on oeis.org

1, 3, 1, 5, 3, 25, 11, 5, 13, 13, 9, 7, 3, 9, 17, 7, 29, 25, 71, 49, 7, 9, 7, 9, 11, 39, 7, 25, 107, 3, 67, 59, 49, 89, 67, 29, 113, 5, 33, 7, 19, 53, 3, 5, 47, 121, 39, 7, 407, 25, 7, 215, 29, 23, 89, 5, 33, 25, 113, 45, 49, 109, 53, 17, 109, 311, 91, 145, 43

Offset: 1

Michel Lagneau, Nov 24 2011

nonn

Cf. A191617, A191618, A191619, A191620, A200921, A200922, A200923.

Table[k = 0; While[!PrimeQ[(3^n + k)*3^n + 1], k++]; k, {n, 72}]

A200921 Least k such that (3^n + k)*3^n - 1 is a prime number.

Original entry on oeis.org

1, 1, 3, 3, 13, 9, 15, 3, 1, 13, 29, 1, 5, 53, 25, 9, 23, 1, 69, 13, 3, 3, 17, 1, 5, 117, 5, 13, 45, 51, 3, 11, 31, 73, 49, 43, 11, 83, 93, 277, 171, 383, 39, 11, 3, 31, 55, 61, 61, 13, 73, 107, 65, 137, 53, 39, 467, 53, 233, 277, 17, 53, 109, 177, 151, 97, 13

Offset: 1

Michel Lagneau, Nov 24 2011

nonn

Cf. A191617, A191618, A191619, A191620, A200920, A200922, A200923.

Table[k = 0; While[!PrimeQ[(3^n + k)*3^n - 1], k++]; k, {n, 72}]
lk[n_]:=Module[{c=3^n,k=0},While[!PrimeQ[(c+k)c-1],k++];k]; Array[lk,70] (* Harvey P. Dale, Aug 19 2015 *)

A200922 Least k such that (3^n - k)*3^n - 1 is a prime number.

Original entry on oeis.org

1, 1, 1, 3, 7, 1, 1, 13, 17, 17, 7, 43, 25, 3, 41, 29, 57, 11, 21, 1, 25, 29, 17, 27, 15, 7, 11, 63, 15, 237, 73, 21, 43, 229, 1, 1, 73, 3, 253, 63, 7, 179, 3, 289, 97, 157, 7, 59, 95, 237, 33, 47, 3, 31, 43, 141, 157, 63, 137, 101, 387, 109, 157, 27, 29, 37

Offset: 1

Michel Lagneau, Nov 24 2011

nonn

Cf. A191617, A191618, A191619, A191620,A200920, A200921, A200923.

Table[k = 0; While[!PrimeQ[(3^n - k)*3^n - 1], k++]; k, {n, 72}]

A200923 Least k such that (3^n - k)*3^n + 1 is a prime number.

Original entry on oeis.org

1, 1, 7, 1, 3, 1, 7, 5, 9, 29, 19, 1, 7, 49, 49, 9, 23, 1, 3, 29, 53, 39, 41, 35, 7, 5, 51, 33, 3, 81, 83, 15, 21, 31, 61, 69, 67, 87, 27, 5, 19, 55, 153, 35, 99, 31, 23, 49, 47, 95, 3, 115, 89, 55, 23, 61, 139, 49, 87, 5, 153, 87, 269, 113, 67, 207, 41, 33

Offset: 1

Michel Lagneau, Nov 24 2011

nonn

Cf. A191617, A191618, A191619, A191620, A200920, A200921, A200922.

Table[k = 0; While[!PrimeQ[(3^n - k)*3^n + 1], k++]; k, {n, 72}]
lk[n_]:=Module[{c=3^n,k=1},While[!PrimeQ[(c-k)*c+1],k++];k]; Array[lk,70] (* Harvey P. Dale, Jul 15 2017 *)

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

Original entry on oeis.org

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

A201133 Least k such that (5^n + k)*5^n + 1 is a prime number.

Original entry on oeis.org

1, 3, 3, 23, 7, 3, 3, 5, 7, 59, 15, 35, 45, 23, 3, 5, 45, 57, 15, 5, 49, 17, 87, 105, 45, 5, 21, 45, 43, 129, 3, 243, 57, 39, 57, 23, 21, 53, 25, 77, 15, 95, 87, 137, 33, 21, 51, 113, 49, 27, 7, 371, 45, 417, 51, 275, 253, 87, 75, 473, 109, 35, 229, 237, 253

Offset: 1

Michel Lagneau, Nov 27 2011

nonn

Cf. A191617, A200920.

Table[k = 0; While[!PrimeQ[(5^n + k)*5^n + 1], k++]; k, {n, 85}]

cp's OEIS Frontend

A191620 Least k such that (2^n-k)*2^n - 1 is a prime number.

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A191618 Least a such that (2^n+a)*2^n - 1 is prime.

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A191619 Least a such that (2^n-a)*2^n + 1 is a prime number.

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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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A200920 Least k such that (3^n+ k)*3^n + 1 is a prime number.

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A200921 Least k such that (3^n + k)*3^n - 1 is a prime number.

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A200922 Least k such that (3^n - k)*3^n - 1 is a prime number.

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A200923 Least k such that (3^n - k)*3^n + 1 is a prime number.

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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.

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A201133 Least k such that (5^n + k)*5^n + 1 is a prime number.

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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.

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