A094076 -id:A094076 - cp's OEIS frontend

A057732 Numbers k such that 2^k + 3 is prime.

1, 2, 3, 4, 6, 7, 12, 15, 16, 18, 28, 30, 55, 67, 84, 228, 390, 784, 1110, 1704, 2008, 2139, 2191, 2367, 2370, 4002, 4060, 4062, 4552, 5547, 8739, 17187, 17220, 17934, 20724, 22732, 25927, 31854, 33028, 35754, 38244, 39796, 40347, 55456, 58312, 122550, 205962, 235326, 363120, 479844, 685578, 742452, 1213815, 1434400, 1594947, 1875552, 1940812, 2205444

Offset: 1

G. L. Honaker, Jr., Oct 29 2000

Some of the larger entries may only correspond to probable primes.

A number k is in this sequence iff A062709(k) is in A057733; this is the case iff A257273(k) is in A125246. - M. F. Hasler, Apr 27 2015

			For k = 6, 2^6 + 3 = 67 is prime.
For k = 28, 2^28 + 3 = 268435459 is prime.

Mike Oakes, posting to primenumbers(AT)yahoogroups.com on Jul 08 2001

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+3, PRP Top Records.

Cf. A050414, A057733, A062709, A094076, A125246, A257273.

Cf. A019434 (primes 2^k+1), this sequence (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+3)]; // Vincenzo Librandi, Apr 27 2015

Select[Range[10000], PrimeQ[2^# + 3] &] (* Vincenzo Librandi, Apr 27 2015 *)

for(n=1, 2200, if(isprime(2^n+3), print1(n, ", ")));

for (n=1, 2, if (isprime(2^n+3), print1(n, ", "))); for(n=3, 100000, N=2^n+3 ; S=(N-5)/2 ; x=S ; for(j=1, n-1, x=Mod(x^2-2, N)) ; if(x==S , print1(n, ", "))) \\ produces terms corresponding to probable primes, see formula; Tony Reix, Aug 27 2015

Here is an LLT-like algorithm, using a cycle of the digraph x^2-2 modulo N, that finds terms of this sequence generating a PRP (PRobable Prime) of A057733 numbers: N=2^k+3; S0=(N-5)/2; s(0)=S0; s(i+1)=s(i)^2-2 modulo N; if s(k-1) == S0 then N is prime. - Tony Reix, Aug 27 2015

More terms from Jason Earls, Jul 18 2001 and Mike Oakes, Jul 28 2001
a(47)-a(50) from Donovan Johnson 2006, verified by Paul Bourdelais, Mar 22 2012
a(51) is a probable prime based on trial factoring to 1E9 and PRP testing base 3,5,7 (PFGW v3.3.1). Discovered by Paul Bourdelais, Apr 09 2012
a(52)-a(54) from Paul Bourdelais, Jun 18 2019
a(55) from Paul Bourdelais, Jul 16 2019
a(56) from Paul Bourdelais, Apr 22 2020
a(57) from Paul Bourdelais, Jun 12 2020
a(58) from Paul Bourdelais, Aug 04 2020

A057196 Numbers k such that 2^k + 9 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 18, 23, 30, 37, 47, 57, 66, 82, 95, 119, 175, 263, 295, 317, 319, 327, 670, 697, 886, 1342, 1717, 1855, 2394, 2710, 3229, 3253, 3749, 4375, 4494, 4557, 5278, 5567, 9327, 10129, 12727, 13615, 14893, 16473, 23639, 40053, 44399, 50335, 80949

Offset: 1

Robert G. Wilson v, Sep 15 2000

nonn

Some of the larger terms are only probable primes.
For these numbers k, 2^(k-1)*(2^k+9) has deficiency 10 (see A101223). - M. F. Hasler, Jul 18 2016
The terms a(48)-a(51) were found by Mike Oakes, a(52) found by Gary Barnes, and a(53-56) found by Lelio R Paula (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 01 2023

			For k = 10, 2^10 + 9 = 1033 is prime.
For k = 30, 2^30 + 9 = 1073741833 is prime.

Robert Price, Table of n, a(n) for n = 1..56
Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+9, PRP Top Records.

Cf. A094076, A101223, A104070 (primes of the form 2^k+9). [Klaus Brockhaus, Mar 14 2009]

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), this sequence (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23). [M. F. Hasler, Jul 18 2016]

Do[ If[ PrimeQ[ 2^n +9 ], Print[n]], { n, 1, 15000 }]

for(n=1, oo, ispseudoprime(2^n+9)&&print1(n", ")) \\ M. F. Hasler, Jul 18 2016

a(48)-a(51) from Mike Oakes, Aug 17 2001

Edited by T. D. Noe, Oct 30 2008

A059242 Numbers k such that 2^k + 5 is prime.

Original entry on oeis.org

1, 3, 5, 11, 47, 53, 141, 143, 191, 273, 341, 16541, 34001, 34763, 42167, 193965, 282203

Offset: 1

Tony Davie (ad(AT)dcs.st-and.ac.uk), Jan 21 2001

The subsequence of primes starts 3, 5, 11, 47, 53, 191, ... - Vincenzo Librandi, Aug 07 2010
For k in this sequence, 2^(k-1)*(2^k+5) is in A141548: numbers of deficiency 6. - M. F. Hasler, Apr 23 2015
a(18) > 5*10^5. - Robert Price, Aug 23 2015
a(18) > 6*10^5. - Tyler NeSmith, Jan 18 2021
All terms are odd - Elmo R. Oliveira, Dec 01 2023

			2^3 + 5 = 13 is prime, but 2^4 + 5 = 21 is not.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+5, PRP Top Records.

Cf. A094076, A141548.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), this sequence (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

Select[Range[20000],PrimeQ[2^#+5]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=ispseudoprime(2^n+5) \\ M. F. Hasler, Apr 23 2015

More terms from Santi Spadaro, Oct 04 2002
a(12) from Hans Havermann, Oct 07 2002
a(13)-a(15) from Charles R Greathouse IV, Oct 07 2011
a(16)-a(17) from Robert Price, Dec 06 2013

A057197 Numbers k such that 2^k + 15 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 22, 23, 26, 30, 32, 40, 42, 46, 61, 72, 76, 155, 180, 198, 203, 310, 328, 342, 508, 510, 515, 546, 808, 1563, 2772, 3882, 3940, 4840, 7518, 11118, 11552, 11733, 12738, 12858, 17421, 44122, 64660, 163560, 172455, 180496, 325866, 481840, 1009168

Offset: 1

Robert G. Wilson v, Sep 15 2000

nonn

a(55) > 5*10^5. - Robert Price, Sep 14 2015

For these numbers k, 2^(k-1)*(2^k+15) has deficiency 16 (see A125248). - M. F. Hasler, Jul 18 2016

			For k = 5, 2^5 + 15 = 47 is prime.
For k = 15, 2^15 + 15 = 32783 is prime.

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+15, PRP Top Records.

Cf. A094076, A125248.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), this sequence (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

[n: n in [0..1500] | IsPrime(2^n+15)]; // Vincenzo Librandi, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 15 ], Print[n]], { n, 1, 12422 }]
Select[Range[15000], PrimeQ[2^# + 15] &] (* Vincenzo Librandi, Aug 28 2015 *)

for(n=1,oo,ispseudoprime(2^n+15)&&print1(n",")) \\ M. F. Hasler, Jul 18 2016

a(45)-a(53) from Robert Price, Dec 06 2013
a(54) from Robert Price, Sep 14 2015
a(55) from Stefano Morozzi, added by Elmo R. Oliveira, Dec 11 2023

A102633 Numbers k such that 2^k + 11 is prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 15, 23, 29, 31, 55, 71, 77, 297, 573, 1301, 1555, 1661, 4937, 5579, 6191, 6847, 6959, 19985, 26285, 47093, 74167, 149039, 175137, 210545, 240295, 306153, 326585, 345547

Offset: 1

Lei Zhou, Jan 20 2005

a(34) > 5*10^5. - Robert Price, Aug 26 2015

For numbers k in this sequence, 2^(k-1)*(2^k+11) has deficiency 12 (see A141549). All terms are odd since 4^n+11 == 1+2 == 0 (mod 3). - M. F. Hasler, Jul 18 2016

			k = 1: 2^1 + 11 = 13 is prime.
k = 3: 2^3 + 11 = 19 is prime.
k = 2: 2^2 + 11 = 15 is not prime.

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+11, PRP Top Records.
Lei Zhou, Between 2^n and primes. [broken link]

Cf. A094076, A141549.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), this sequence (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

Do[ If[ PrimeQ[2^n + 11], Print[n]], {n, 15250}] (* Robert G. Wilson v, Jan 21 2005 *)

for(n=1,9e9,ispseudoprime(2^n+11)&&print1(n",")) \\ M. F. Hasler, Jul 18 2016

a(18)-a(22) from Robert G. Wilson v, Jan 21 2005
a(23)-a(33) from Robert Price, Dec 06 2013
Edited by M. F. Hasler, Jul 18 2016

A057201 Numbers k such that 2^k + 21 is prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 8, 11, 15, 16, 19, 44, 48, 51, 52, 61, 163, 196, 456, 492, 911, 997, 1616, 1631, 1647, 1803, 1899, 3112, 3584, 3956, 6848, 7023, 9535, 16657, 27035, 33843, 36551, 38859, 81485, 107287, 131383, 139476, 158497, 210061, 216752, 339168, 341355, 376731, 1173095

Offset: 1

Robert G. Wilson v, Sep 16 2000

nonn

a(48) > 5*10^5. - Robert Price, Sep 17 2015

			k = 15, 2^15 + 21 = 32789 is prime.
k = 16, 2^16 + 21 = 65557 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+21, PRP Top Records.

Cf. A094076.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), this sequence (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+21)]; // Vincenzo Librandi, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 21 ], Print[ n ] ], {n, 1, 4000} ]
Select[Range[10000], PrimeQ[2^# + 21] &] (* Vincenzo Librandi, Aug 28 2015 *)

is(n)=isprime(2^n+21) \\ Charles R Greathouse IV, Feb 17 2017

a(30)-a(47) from Robert Price, Dec 06 2013

a(48) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 25 2023

A057203 Numbers k such that 2^k + 23 is prime.

Original entry on oeis.org

3, 7, 39, 79, 359, 451, 1031, 1039, 11311, 30227, 47599, 55731, 307099, 351831, 418851

Offset: 1

Robert G. Wilson v, Sep 16 2000

a(16) > 5*10^5. - Robert Price, Sep 06 2015

All terms are odd. - Elmo R. Oliveira, Dec 01 2023

			For k = 39, 2^39 + 23 = 549755813911 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+23, PRP Top Records.

Cf. A094076.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196(2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), this sequence (2^k+23).

Do[ If[ PrimeQ[2^n + 23], Print[ n ]], {n, 1, 5000} ]

is(n)=isprime(2^n+23) \\ Charles R Greathouse IV, Feb 17 2017

a(9)-a(15) from Robert Price, Sep 06 2015

A067760 a(n) is the least positive k such that (2n+1) + 2^k is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 4, 2, 1, 2, 1, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 8, 2, 1, 2, 1, 1, 4, 2, 1, 2, 1, 7, 2, 1, 3, 4, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 7, 4, 5, 3, 4, 2, 1, 2, 1, 3, 2, 1, 1, 10, 3, 3, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 2, 1, 1, 4, 3, 3, 2, 1, 1, 2, 1, 1, 6, 5, 3, 6

Offset: 0

Don Reble, Feb 05 2002

nonn

From Phil Moore (moorep(AT)lanecc.edu), Dec 14 2009: (Start)
It is known that a(39278) = 0, since no such prime exists for the Sierpiński number 78557 (cf. A076336).
It has recently been discovered that 2131+2^4583176 and 41693+2^5146295 are probable primes, so a(1065) is probably 4583176 and a(20846) is probably 5146295.
At present, the only odd value less than 78557 for which no prime or strong probable prime of the form t+2^k is known is t = 40291, so a(20145) is completely unknown. In addition, for 25 values of t < 78557, only strong probable primes are known. (End)
The last case was resolved in 2011 when the probable prime 40291+2^9092392 was found as a part of a distributed project "Five or Bust". See links. - Jeppe Stig Nielsen, Mar 29 2019

			a(15)=4 because (2*15+1)+2^k is composite for k=1,2,3 and prime for k=4.

Jinyuan Wang, Table of n, a(n) for n = 0..39278 (terms 0..1064 from T. D. Noe, terms 1065..3000 from Richard N. Smith).
Mersenne Forum, Five or Bust
Carlos Rivera, Puzzle 167. Primes m + 2^j & m - 2^j, The Prime Puzzles and Problems Connection.

Cf. A016014, A050412, A066081, A033919, A094076, A076336, A260350, A263874, A263875 (records).

a(n) = {my(k=1); while (! isprime((2*n+1)+2^k), k++); k;} \\ Michel Marcus, Feb 26 2018

A065381 Primes not of the form p + 2^k, p prime and k >= 0.

Original entry on oeis.org

2, 127, 149, 251, 331, 337, 373, 509, 599, 701, 757, 809, 877, 907, 977, 997, 1019, 1087, 1259, 1549, 1597, 1619, 1657, 1759, 1777, 1783, 1867, 1973, 2203, 2213, 2293, 2377, 2503, 2579, 2683, 2789, 2843, 2879, 2909, 2999, 3119, 3163, 3181, 3187, 3299

Offset: 1

Reinhard Zumkeller, Nov 03 2001

nonn

Sequence is infinite. For example, Pollack shows that numbers which are 1260327937 mod 2863311360 are not of the form p + 2^k for any prime p and k >= 0, and there are infinitely many primes in this congruence class by Dirichlet's theorem. - Charles R Greathouse IV, Jul 20 2014

			127 is a prime, 127-2^0 through 127-2^6 are all nonprimes.

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff. [?Broken link]
P. Pollack, Not Always Buried Deep: Selections from Analytic and Combinatorial Number Theory, p. 193, ex. 5.1.6, p. 216ff.
Lei Zhou, Between 2^n and primes.

Equals A000040 minus A065380.
Cf. A010051, A006285, A102630, A094076, A156695.
Cf. A098237.

a065381 n = a065381_list !! (n-1)
a065381_list = filter f a000040_list where
   f p = all ((== 0) . a010051 . (p -)) $ takeWhile (<= p) a000079_list
-- Reinhard Zumkeller, Nov 24 2011

fQ[n_] := Block[{k = Floor[Log[2, n]], p = n}, While[k > -1 && ! PrimeQ[p - 2^k], k--]; If[k > 0, True, False]]; Drop[Select[Prime[Range[536]], ! fQ[#] &], {2}] (* Robert G. Wilson v, Feb 10 2005; corrected by Arkadiusz Wesolowski, May 05 2012 *)

is(p)=my(k=1);while(kp,return(isprime(p)));0 \\ Charles R Greathouse IV, Jul 20 2014

A078687(A049084(a(n))) = 0; subsequence of A118958. - Reinhard Zumkeller, May 07 2006

Link and cross-reference fixed by Charles R Greathouse IV, Nov 09 2008

A157007 Numbers k such that 2^k + 27 is prime.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 14, 16, 40, 41, 44, 86, 110, 125, 133, 134, 145, 154, 184, 194, 301, 308, 320, 685, 1001, 1066, 1496, 1633, 2005, 2864, 3241, 6286, 11585, 12854, 16514, 16540, 19246, 24538, 28705, 57644, 65366, 85276, 89113, 194854, 266680, 376790, 478088

Offset: 1

Edwin Dyke (ed.dyke(AT)btinternet.com), Feb 20 2009

nonn

a(49) > 5*10^5. - Robert Price, Nov 06 2015

			For k = 1, 2^1 + 27 = 29.
For k = 2, 2^2 + 27 = 31.
For k = 4, 2^4 + 27 = 43.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+27, PRP Top Records.

Cf. A057733, A094076, A104071, A123250.

Cf. A019434 (primes 2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (2^k+13), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23), A157006 (2^k+25), this sequence (2^k+27), A156982 (2^k+29), A247952 (2^k+31), A247953 (2^k+33), A220077 (2^k+35).

[n: n in [0..1000] | IsPrime(2^n+27)]; // Vincenzo Librandi, Oct 05 2015

Delete[Union[Table[If[PrimeQ[2^n + 27], n, 0], {n, 1, 2000}]], 1]
Select[Range[5000],PrimeQ[2^#+27]&] (* Harvey P. Dale, Mar 24 2011 *)

for(n=1, 1e3, if(isprime(2^n+3^3), print1(n", "))) \\ Altug Alkan, Oct 04 2015

More terms from Harvey P. Dale, Mar 24 2011
a(33)-a(42) from Robert Price, Oct 04 2015
a(43)-a(47) discovered by Henri Lifchitz and Lelio R Paula from Lifchitz link by Robert Price, Oct 04 2015
a(48) from Robert Price, Nov 06 2015

cp's OEIS Frontend

A057732 Numbers k such that 2^k + 3 is prime.

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A057196 Numbers k such that 2^k + 9 is prime.

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A059242 Numbers k such that 2^k + 5 is prime.

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A057197 Numbers k such that 2^k + 15 is prime.

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A102633 Numbers k such that 2^k + 11 is prime.

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A057201 Numbers k such that 2^k + 21 is prime.

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A057203 Numbers k such that 2^k + 23 is prime.