A059242 -id:A059242 - 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

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

A057200 Numbers k such that 2^k + 17 is prime.

Original entry on oeis.org

1, 13, 21, 33, 81, 129, 285, 297, 769, 3381, 4441, 7065, 77121, 133437, 184189, 191745, 1279921

Offset: 1

Robert G. Wilson v, Sep 16 2000

a(17) > 5*10^5. - Robert Price, Oct 05 2015
For numbers k in this sequence, 2^(k-1)*(2^k+17) has deficiency 18 (see A223608). - M. F. Hasler, Jul 18 2016
All terms are odd. - Elmo R. Oliveira, Nov 19 2023

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+17, PRP Top Records.

Cf. A223608.

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), this sequence (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

[n: n in [0..1000] | IsPrime(2^n+17)]; // Vincenzo Librandi, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 17 ], Print[ n ]], {n, 0, 11811} ]
Select[Range[10000], PrimeQ[2^# + 17] &] (* Vincenzo Librandi, Aug 28 2015 *)

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

a(13)-a(16) from Robert Price, Aug 24 2015
Edited by M. F. Hasler, Jul 18 2016
a(17) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 19 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

A123250 Primes of the form 2^k + 5.

Original entry on oeis.org

7, 13, 37, 2053, 140737488355333, 9007199254740997, 2787593149816327892691964784081045188247557, 11150372599265311570767859136324180752990213, 3138550867693340381917894711603833208051177722232017256453

Offset: 1

Cino Hilliard, Oct 08 2006

nonn

A059242 is the main entry for this sequence.

If 2^n + 5 is prime then n is odd. Proof: Lemma 1: a^n+b^n = (a+b)(a^n-1 - a^(n-2)b + ... + b^(n-1)) 2^n + 5 = 2*(2^(n-1)+1) + 3. Then if n is even, n-1 is odd and by Lemma 1, 2+1 divides 2*(2^(n-1)+1) and thus divides 2^n+5 so it cannot be prime.

Cf. A000040, A059242.

Select[Table[2^k+5,{k,200}],PrimeQ] (* James C. McMahon, Nov 19 2024 *)

g(n,p) = for(k=1,n,y=p+2^k;if(isprime(y),print1(y",")))

a(n) = 2^A059242(n) + 5. - Elmo R. Oliveira, Nov 08 2023

a(9) from James C. McMahon, Nov 19 2024

A141548 Numbers n whose deficiency is 6.

Original entry on oeis.org

7, 15, 52, 315, 592, 1155, 2102272, 815634435

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2008

a(9) > 10^12. - Donovan Johnson, Dec 08 2011
a(9) > 10^13. - Giovanni Resta, Mar 29 2013
a(9) > 10^18. - Hiroaki Yamanouchi, Aug 21 2018
For all k in A059242, the number m = 2^(k-1)*(2^k+5) is in this sequence. This yields further terms 2^46*(2^47+5), 2^52*(2^53+5), 2^140*(2^141+5), ... All even terms known so far and the initial 7 = 2^0*(2^1+5) are of this form. All odd terms beyond a(2) are of the form a(n) = a(k)*p*q, k < n. We have proved that there is no further term of this form with the a(k) given so far. - M. F. Hasler, Apr 23 2015
A term n of this sequence multiplied by a prime p not dividing it is abundant if and only if p < sigma(n)/6 = n/3-1. For the even terms 592 and 2102272, there is such a prime near this limit (191 resp. 693571) such that n*p is a primitive weird number, cf. A002975. For a(3)=52, the largest such prime, 11, is already too small. Odd weird numbers do not exist within these limits. - M. F. Hasler, Jul 19 2016
Any term x of this sequence can be combined with any term y of A087167 to satisfy the property (sigma(x)+sigma(y))/(x+y) = 2, which is a necessary (but not sufficient) condition for two numbers to be amicable. - Timothy L. Tiffin, Sep 13 2016

			a(1) = 7, since 2*7 - sigma(7) = 14 - 8 = 6. - _Timothy L. Tiffin_, Sep 13 2016

Gianluca Amato, Maximilian Hasler, Giuseppe Melfi, Maurizio Parton, Primitive weird numbers having more than three distinct prime factors, Riv. Mat. Univ. Parma, 7(1), (2016), 153-163, arXiv:1803.00324 [math.NT], 2018.

Cf. A087485 (odd terms).
Cf. A000203, A033880, A005100; A191363 (deficiency 2), A125246 (deficiency 4), A141548 (deficiency 6), A125247 (deficiency 8), A101223 (deficiency 10), A141549 (deficiency 12), A141550 (deficiency 14), A125248 (deficiency 16), A223608 (deficiency 18), A223607 (deficiency 20).
Cf. A087167 (abundance 6).

[n: n in [1..9*10^6] | (SumOfDivisors(n)-2*n) eq -6]; // Vincenzo Librandi, Sep 14 2016

lst={};Do[If[n==Plus@@Divisors[n]-n+6,AppendTo[lst,n]],{n,10^4}];Print[lst];
Select[Range[1, 10^8], DivisorSigma[1, #] - 2 # == - 6 &] (* Vincenzo Librandi, Sep 14 2016 *)

is(n)=sigma(n)==2*n-6 \\ Charles R Greathouse IV, Apr 23 2015, corrected by M. F. Hasler, Jul 18 2016

a(8) from Donovan Johnson, Dec 08 2011

A102634 Numbers k such that 2^k + 13 is prime.

Original entry on oeis.org

2, 4, 8, 20, 38, 64, 80, 292, 1132, 4108, 19934, 125278, 175628, 282184

Offset: 1

Lei Zhou, Jan 20 2005

If k is odd, then 2^k + 13 is divisible by 3. - Robert G. Wilson v, Jan 24 2005
a(15) > 5*10^5. - Robert Price, Aug 15 2015
For k in this sequence, the number 2^(k-1)*(2^k+13) has deficiency 14, cf. A141550. - M. F. Hasler, Jul 18 2016

			2^2+13 = 17 is prime.
2^4+13 = 29 is prime.
2^3+13 = 21 is not prime.

H. Lifchitz and R. Lifchitz (Editors), PRP Top Records, of the form 2^n+13.
Lei Zhou, Between 2^n and primes [broken link].

Cf. A019434 (2^k+1), A057732 (2^k+3), A059242 (2^k+5), A057195 (2^k+7), A057196 (2^k+9), A102633 (2^k+11), A102634 (this), A057197 (2^k+15), A057200 (2^k+17), A057221 (2^k+19), A057201 (2^k+21), A057203 (2^k+23).

Do[m = n; If[PrimeQ[2^n + 13], Print[n]], {n, 2, 19125, 2}] (* Robert G. Wilson v, Jan 24 2005 *)

first(m)=my(v=vector(m),r=1);for(i=1,m,while(!isprime(2^r + 13),r++);v[i]=r;r++);v; \\ Anders Hellström, Aug 15 2015

a(n) = 2*A253772(n). - Elmo R. Oliveira, Nov 12 2023

a(10) from Robert G. Wilson v, Jan 24 2005

a(11)-a(14) from Robert Price, Aug 15 2015

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

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

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A057200 Numbers k such that 2^k + 17 is prime.

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

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A123250 Primes of the form 2^k + 5.

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A141548 Numbers n whose deficiency is 6.

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