A194591 -id:A194591 - cp's OEIS frontend

A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.

2, 2, 11, 13, 17, 23, 53, 29, 67, 37, 41, 47, 101, 53, 59, 61, 67, 71, 73, 79, 83, 173, 89, 751, 97, 101, 107, 109, 113, 1889, 487, 127, 131, 269, 137, 283, 293, 149, 307, 157, 163, 167, 1361, 173, 179, 181, 373, 191, 193, 197, 809, 823, 211, 857, 6977, 223, 227

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

nonn

Bisection of A194603.

Primes arising from A194636 (or 0 if no such prime exists).

			For n=4, 7*2^0-1 and 7*2^0+1 are composite, but 7*2^1-1=13 is prime, so a(4)=13.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194637, A194639.

Cf. A038699, A050921, A076335, A180247.

Table[n = 2*n - 1; k = 0; While[! PrimeQ[a = n*2^k - 1] && ! PrimeQ[a = n*2^k + 1], k++]; a, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)

A194607 Record values in A194606.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 11, 17, 20, 28, 70, 99, 150, 726, 7431, 22394, 85461, 191207

Offset: 1

Arkadiusz Wesolowski, Aug 30 2011

The index sequence of this one is 1, 3, 6, 15, 17, 29, 53, 115, 186, 220, 229, 1886, 5344, 5736, 66774, 1087403, 14747671, 158018119.

a(17) was found in 2000 by Wilfrid Keller and a(18) was found in 2003 by Patrick De Geest.

			A194606(53) = 11 since A194606(115) = 17 is the next record value.

Wilfrid Keller, personal communication, 2010.

Chris K. Caldwell, The List of Largest Known Primes, 270704167 * 2^85461 - 1
Chris K. Caldwell, The List of Largest Known Primes, 3296757029 * 2^191207 + 1
Carlos Rivera, Problem 30

Cf. A194591, A194600, A194603, A194606, A194608, A194635, A194636, A194637, A194638, A194639.

Cf. A103963, A103964, A076335, A180247.

l = -1; Flatten[Table[p = Prime[n]; k = 0; While[! PrimeQ[p*2^k - 1] && ! PrimeQ[p*2^k + 1], k++]; If[k > l, l = k, {}], {n, 10^4}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

A194637 Record values in A194636.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 11, 18, 20, 28, 70, 106, 150, 726, 2906, 7431, 14073, 22394, 41422, 82587, 85461, 356981

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

The index sequence is 1, 3, 7, 24, 30, 55, 121, 168, 555, 687, 724, 7447, 26134, 28272, 324802, 419221, 4420051, 8467881, 50302257, 59186640, 135352084, 677738616, ... given by formula (A194639(n)+1)/2.

			A194636(55) = 6 since A194636(121) = 11 is the next record value.

Wilfrid Keller, personal communication, 2010.

Chris K. Caldwell, The List of Largest Known Primes, 1355477231 * 2^356981 + 1
Carlos Rivera, Problem 30

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194636, A194638, A194639.

Cf. A103963, A103964, A076335, A180247.

l = -1; Flatten[Table[n = 2*n - 1; k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; If[k > l, l = k, {}], {n, 10^5}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

a(n) = A194591(A194639(n)) = A194636((A194639(n)+1)/2).

a(22) was found in 2002 by Wilfrid Keller.

A253176 Least k>=0 such that both 3n2^k+1 and 3n2^k-1 are primes, or -1 if no such k exists.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 3, 2, 0, 6, 1, 3, 0, 2, 2, 1, 1, 2, 0, 14, 5, 1, 0, 1, 2, 5, 5, 2, 1, 4, 1, 1, 0, 2, 0

Offset: 1

Eric Chen, Mar 16 2015

If n*2^k+1 and n*2^k-1 are both primes, then n must be divisible by 3.
a(79) = -1, since 237*2^k+1 or 237*2^k-1 must divisible by 5, 7, 13, 17, or 241. Similarly, a(269) = -1 (cover: {5, 7, 13, 19, 37, 73}), a(1527) = -1 (cover: {5, 7, 13, 17, 241}).
Conjecture: if n < 79, then a(n) >= 0.
a(38) - a(40) = {1, 4, 1}, a(42) - a(50) = {13, 3, 4, 1, 0, 1, 3, 44, 0}, a(52) = 1, a(54) - a(56) = {4, 2, 4}, a(58) - a(60) = {1, 12, 0}, a(n) is currently unknown for n = {37, 41, 51, 53, 57, ...}
a(37), if it exists, is > 160000.

			a(11) = 6 since 33*2^n+1 and 33*2^n-1 are not both primes for all 0 <= n <= 5, but they are both primes for n = 6.

Ray Ballinger and Wilfrid Keller, Primes of the form k*2^n+1 for k up to 300
Ray Ballinger and Wilfrid Keller, Primes of the form k*2^n-1 for k up to 300
Eric Chen, Table of n, a(n) for n = 1..1000 (-1 if this term is unknown or does not exist)
Carlos Rivera, Problem 49. Sierpinski-like numbers

Cf. A040076, A040081, A194591.

Table[k = 0; While[! PrimeQ[3*n*2^k + 1] || ! PrimeQ[3*n*2^k - 1], k++]; k, {n, 60}]

a(n) = for(k=0, 2^24, if(ispseudoprime(3*n*2^k+1) && ispseudoprime(3*n*2^k-1), return(k)))

If a(n) > 0, then a(2n) = a(n) - 1.

cp's OEIS Frontend

A194638 Smallest prime either of the form (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1, k >= 0, or 0 if no such prime exists.

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A194607 Record values in A194606.

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A194637 Record values in A194636.

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A253176 Least k>=0 such that both 3n*2^k+1 and 3n*2^k-1 are primes, or -1 if no such k exists.

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A194638 Smallest prime either of the form (2n-1)2^k - 1 or (2n-1)2^k + 1, k >= 0, or 0 if no such prime exists.

A253176 Least k>=0 such that both 3n2^k+1 and 3n2^k-1 are primes, or -1 if no such k exists.