A076335 -id:A076335 - cp's OEIS frontend

A194636 Least k >= 0 such that (2n-1)2^k - 1 or (2n-1)2^k + 1 is prime, or -1 if no such value exists.

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

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

sign

Bisection of A194591: a(n) = A194591(2*n-1).

A194637 gives the record values.

			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)=1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Brier Number

Cf. A194591, A194600, A194603, A194606, A194607, A194608, A194635, A194637, A194638, A194639.

Cf. A040081, A040076, A076335, A180247.

Table[n = 2*n - 1; k = 0; While[! PrimeQ[n*2^k - 1] && ! PrimeQ[n*2^k + 1], k++]; k, {n, 100}] (* Arkadiusz Wesolowski, Sep 04 2011 *)
p[n_]:=Module[{c=2n-1,k=0},While[!Or@@PrimeQ[c*2^k+{1,-1}],k++];k]; Array[ p,90] (* Harvey P. Dale, Mar 08 2013 *)

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.

Original entry on oeis.org

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 *)

A194639 Indices of records in A194591 when it is restricted to odd indices.

Original entry on oeis.org

1, 5, 13, 47, 59, 109, 241, 335, 1109, 1373, 1447, 14893, 52267, 56543, 649603, 838441, 8840101, 16935761, 100604513, 118373279, 270704167, 1355477231

Offset: 1

Arkadiusz Wesolowski, Aug 31 2011

Integers for which the smallest k in A194591 such that (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1 is prime (A194638) increases.

A194637 gives the record values of A194636.

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, A194637, A194638.
Cf. A064699, A076335, A180247, A217892.
Cf. A217892 (indices of records of unrestricted A194591)

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, {}], {n, 10^5}]] (* Arkadiusz Wesolowski, Sep 04 2011 *)

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

Definition corrected by Max Alekseyev and Farideh Firoozbakht, Oct 16 2014

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.

A255967 Odd numbers m that are neither of the form p + 2^k nor of the form p - 2^k with 2^k < m, k >= 1, and p prime.

Original entry on oeis.org

1, 1973, 3181, 3967, 4889, 5617, 7747, 7913, 8363, 8587, 8923, 11437, 11993, 12517, 13285, 13973, 14101, 14231, 14489, 16117, 16769, 16849, 18391, 18611, 19583, 19819, 21289, 21683, 21701, 21893, 22147, 22817, 22949, 23651, 24943, 25829, 27197, 27437

Offset: 1

Arkadiusz Wesolowski, Mar 12 2015

nonn

Odd numbers m such that for all 2^k < m the numbers m + 2^k and m - 2^k are composite, with k >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A076335.
Subsequence of A006285. Supersequence of A256163.
A153352 gives the primes.

lst:=[]; for n in [1..27437 by 2] do t:=0; k:=0; while 2^k lt n do if IsPrime(n-2^k) or IsPrime(n+2^k) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, n); end if; end for; lst;

q[m_] :=  If[EvenQ[m], False, Module[{pow = 2},While[pow < m && !PrimeQ[m - pow] && !PrimeQ[m + pow], pow *= 2]; pow > m]]; Select[Range[30000], q] (* Amiram Eldar, Jul 19 2025 *)

isok(m) = if(!(m % 2), 0, my(pow = 2); while(pow < m && !isprime(m - pow) && !isprime(m + pow), pow *= 2); pow > m); \\ Amiram Eldar, Jul 19 2025

A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m2^k + 1, and m2^k - 1 are composite, with k >= 1.

Original entry on oeis.org

1, 7913, 8923, 24943, 34009, 35437, 42533, 52783, 60113, 83437, 100727, 105953, 116437, 120521, 126631, 132211, 133241, 137171, 145589, 164729, 172331, 181645, 183671, 192173, 196633, 199513, 203069, 204013, 215113, 215279, 218503, 220523, 253519, 254329, 254587

Offset: 1

Arkadiusz Wesolowski, Mar 17 2015

nonn

Felix Fröhlich, Table of n, a(n) for n = 1..10000

Cf. A006285, A076335.
Subsequence of A255967.
A256237 gives the primes.

lst:=[]; for n in [1..254587 by 2] do t:=0; k:=0; while 2^k lt n do if IsPrime(n-2^k) or IsPrime(n+2^k) or IsPrime(n*2^k-1) or IsPrime(n*2^k+1) then t:=1; break; end if; k+:=1; end while; if IsZero(t) then Append(~lst, n); end if; end for; lst;

q[m_] :=  If[EvenQ[m], False, Module[{pow = 2},While[pow < m && !PrimeQ[m - pow] && !PrimeQ[m + pow] && !PrimeQ[m * pow - 1] && !PrimeQ[m * pow + 1], pow *= 2]; pow > m]]; Select[Range[300000], q] (* Amiram Eldar, Jul 19 2025 *)

for(n=1, 1e6, if(n%2==1, k=0; prim=0; while(2^k < n, if(ispseudoprime(n+2^k) || ispseudoprime(n-2^k) || ispseudoprime(n*2^k+1) || ispseudoprime(n*2^k-1), prim++; break({1})); k++); if(prim==0, print1(n, ", ")))) \\ Felix Fröhlich, Apr 01 2015

A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

Original entry on oeis.org

30666137, 31210219, 52109063, 52504261, 55414847, 55876981, 57816799, 60097043, 63723707, 68748319, 79933129, 87747827, 88486403, 93034073, 104218883, 131873509, 138385817, 152485283, 155269609, 158241023, 165795677, 166441831, 177702619, 197903207

Offset: 1

Arkadiusz Wesolowski, Oct 22 2015

nonn

Odd n such that for all k > 0 the numbers n + 2^k and n - 2^k are nonprimes.

Cf. A006285, A076335, A076336. Subsequence of A255967. A263645 gives the primes.

A006285 INTERSECT A076336.

A364413 Odd numbers m such that for every k >= 1, m*2^k + 1 has a divisor in the set {3, 7, 11, 19, 31, 37, 41, 61, 73, 109, 151, 331}.

Original entry on oeis.org

189035277393779, 212050850472529, 618127765127603, 777947701660121, 1171304921532749, 1358735367828947, 1834310020939021, 2357654372323739, 2638037471052913, 3025664372930897, 3935005074246167, 4688754513654559, 4996748200142999, 5425272498782051, 5455203077891285

Offset: 1

Arkadiusz Wesolowski, Jul 23 2023

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..34560

Cf. A076336, A076335, A180247, A291360, A364412.

For n > 34560, a(n) = a(n-34560) + 10014447295554878022.

A263347 Odd numbers n such that for every k >= 1, n*2^k + 1 has a divisor in the set {3, 5, 13, 17, 97, 241, 257}.

Original entry on oeis.org

37158601, 1017439067, 1242117623, 1554424697, 1905955429, 2727763433, 4512543497, 4798554619, 4954643117, 4988327659, 5367644183, 5660978867, 6107173883, 7173264623, 7425967459, 8365215091, 8776906457, 9013226179, 9095014883, 9787717801, 10466795551

Offset: 1

Arkadiusz Wesolowski, Oct 15 2015

nonn

Cohen and Selfridge showed that this sequence contains infinitely many numbers that are both Sierpiński and Riesel.
What is the smallest term of this sequence that belongs to A076335? Is it the smallest Brier number?
This sequence contains only numbers of the form 30*k + 1, 30*k + 17, 30*k + 19, and 30*k + 23.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..96
Chris Caldwell, The Prime Glossary, Sierpinski number
Fred Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comput. 29 (1975), pp. 79-81.
Carlos Rivera, Problem 29 and Problem 58
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A076335, A263561.
Subsequence of A076336.
A263560 gives the primes.

a(n) = a(n-96) + 39832304070 for n > 96.

cp's OEIS Frontend

A194636 Least k >= 0 such that (2*n-1)*2^k - 1 or (2*n-1)*2^k + 1 is prime, or -1 if no such value exists.

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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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A194639 Indices of records in A194591 when it is restricted to odd indices.

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A194607 Record values in A194606.

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A194637 Record values in A194636.

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A255967 Odd numbers m that are neither of the form p + 2^k nor of the form p - 2^k with 2^k < m, k >= 1, and p prime.

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A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m*2^k + 1, and m*2^k - 1 are composite, with k >= 1.

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A263644 Odd numbers that are neither of the form p + 2^k nor of the form p - 2^k with k > 0, and p prime.

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A194636 Least k >= 0 such that (2n-1)2^k - 1 or (2n-1)2^k + 1 is prime, or -1 if no such value exists.

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.

A256163 Odd numbers m such that for all 2^k < m the numbers m + 2^k, m - 2^k, m2^k + 1, and m2^k - 1 are composite, with k >= 1.