A057203 -id:A057203 - cp's OEIS frontend

A247952 Numbers k such that 2^k + 31 is prime.

4, 12, 36, 540, 844, 1192, 12136, 84280, 128356, 317464, 3018556

Offset: 1

Vincenzo Librandi, Sep 28 2014

Some terms correspond to probable primes. Lifchitz link shows Paul Underwood discovered 84280, and Lelio R Paula found 128356 and 317464 are in the sequence. - Jens Kruse Andersen, Sep 29 2014
a(11) > 5*10^5. - Robert Price, Oct 25 2015
All terms are even. - Elmo R. Oliveira, Nov 25 2023

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+31, PRP Top Records.

Cf. A094076, A262971.

Cf. Numbers k such that 2^k + d is prime: (0,1,2,4,8,16) for d=1; A057732 (d=3), A059242 (d=5), A057195 (d=7), A057196 (d=9), A102633 (d=11), A102634 (d=13), A057197 (d=15), A057200 (d=17), A057221 (d=19), A057201 (d=21), A057203 (d=23), A157006 (d=25), A157007 (d=27), A156982 (d=29), this sequence (d=31), A247953 (d=33), A220077 (d=35).

Magma
```
[n: n in [0..2000]| IsPrime(2^n+31)];
```

Select[Range[0,10000], PrimeQ[2^# + 31] &]

is(n)=ispseudoprime(2^n+31) \\ Charles R Greathouse IV, May 22 2017

a(n) = 2*A262971(n). - Elmo R. Oliveira, Nov 25 2023

12136 and 84280 from Jens Kruse Andersen, Sep 29 2014
a(9)-a(10) (discovered by Lelio R Paula; see the Lifchitz link) added by Robert Price, Oct 04 2015
a(11) discovered by Robert Price, added by Elmo R. Oliveira, Nov 25 2023

A247953 Numbers k such that 2^k + 33 is prime.

Original entry on oeis.org

2, 3, 6, 11, 12, 14, 15, 20, 30, 60, 68, 75, 108, 116, 135, 206, 210, 410, 446, 558, 851, 1482, 1499, 2039, 2051, 4196, 7046, 7155, 8735, 10619, 18420, 20039, 46719, 75348, 179790, 203018, 434246

Offset: 1

Vincenzo Librandi, Sep 28 2014

Some terms correspond to probable primes. Lifchitz link shows the terms 179790 found by Donovan Johnson and 203018 by Lelio R Paula. - Jens Kruse Andersen, Sep 30 2014

a(38) > 5*10^5. - Robert Price, Nov 07 2015

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+33, PRP Top Records.

Cf. A094076, A176926, A247957.

Cf. Numbers k such that 2^k + d is prime: (0,1,2,4,8,16) for d=1; A057732 (d=3), A059242 (d=5), A057195 (d=7), A057196 (d=9), A102633 (d=11), A102634 (d=13), A057197 (d=15), A057200 (d=17), A057221 (d=19), A057201 (d=21), A057203 (d=23), A157006 (d=25), A157007 (d=27), A156982 (d=29), A247952 (d=31), this sequence (d=33), A220077 (d=35).

/* The code gives only the terms up to 851: */ [n: n in [1..1400]| IsPrime( 2^n + 33 )];

A247957:=n->`if`(isprime(2^n+33),n,NULL): seq(A247957(n), n=0..1000); # Wesley Ivan Hurt, Sep 28 2014

Select[Range[10000], PrimeQ[2^# + 33] &]

is(n)=ispseudoprime(2^n+33) \\ Charles R Greathouse IV, Feb 20 2017

a(30)-a(34) from Jens Kruse Andersen, Sep 30 2014
a(35)-a(36) (discovered by Donovan Johnson and Lelio R Paula, respectively; see the Lifchitz link) added by Robert Price, Oct 04 2015
a(37) from Robert Price, Nov 07 2015

A156982 Numbers k such that 2^k + 29 is prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 23, 27, 33, 37, 43, 63, 69, 73, 79, 89, 117, 127, 239, 395, 409, 465, 837, 2543, 10465, 10837, 17005, 19285, 24749, 26473, 29879, 49197, 56673, 67119, 67689, 71007, 109393, 156403, 158757, 181913, 190945, 207865, 222943, 419637

Offset: 1

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

nonn

n cannot be of the form 4m+2 or 4m because 2^(2m+2) + 29 is divisible by 3 and 2^4m + 29 is divisible by 15. - Avik Roy (avik_3.1416(AT)yahoo.co.in), Feb 21 2009

a(47) > 5*10^5. - Robert Price, Oct 25 2015

			For k = 1, 2^1 + 29 = 31.
For k = 3, 2^3 + 29 = 37.

Henri Lifchitz and Renaud Lifchitz, PRP Records, search for 2^n+29

Cf. A019434 (Fermat primes 2^(2^n)+1).

Cf. A057732, A059242, A057195, A057196, A102633, A102634, A057197, A057200, A057221, A057201, A057203, A157006, A157007, A247952, A247953, A220077.

[n: n in [0..1000] | IsPrime(2^n+29)]; // Vincenzo Librandi, Oct 05 2015

Delete[Union[Table[If[PrimeQ[2^n + 29], n, 0], {n, 1, 2600}]], 1]
Select[Range[500000], PrimeQ[2^#+29]&] (* Robert Price, Oct 04 2015 *)

is(n)=ispseudoprime(2^n+29) \\ Charles R Greathouse IV, Jun 06 2017

a(27)-a(38) from Robert Price, Oct 04 2015

a(39)-a(46) discovered by Henri Lifchitz from Lifchitz link by Robert Price, Oct 04 2015

A157006 Numbers k such that 2^k + 25 is prime.

Original entry on oeis.org

2, 4, 6, 8, 10, 20, 22, 34, 70, 92, 112, 118, 236, 250, 378, 438, 570, 654, 800, 1636, 2848, 4948, 5670, 6772, 7494, 8006, 9056, 11038, 16268, 21416, 21738, 33370, 78706, 112130, 126446, 164046, 219250, 236432, 368048, 524154, 530810, 640854, 699740, 746302, 754038, 754376, 931976, 989562

Offset: 1

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

nonn

a(40) > 5*10^5. - Robert Price, Oct 15 2015

Since each term is even (n = 2*k), prime numbers of the form 2^k + 25 (see A104072) also have the form 4^k + 25. Those values of k are given in A204388. - Timothy L. Tiffin, Aug 06 2016

			For k = 2, 2^2 + 25 = 29.
For k = 4, 2^4 + 25 = 41.
For k = 6, 2^6 + 25 = 89.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+25, PRP Top Records.

Cf. A094076, A104072, A123250, A204388.

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), this sequence (2^k+25), A157007 (2^k+27), A156982 (2^k+29), A247952 (2^k+31), A247953 (2^k+33), A220077 (2^k+35).

[n: n in [1..1000] | IsPrime(2^n+25)]; // Vincenzo Librandi, Aug 07 2016

Delete[Union[Table[If[PrimeQ[2^n + 25], n, 0], {n, 1, 1000}]], 1]
Select[Range[0, 10000], PrimeQ[2^# + 25] &] (* Vincenzo Librandi, Aug 07 2016 *)

is(n)=ispseudoprime(2^n+5^2) \\ Charles R Greathouse IV, Feb 20 2017

a(n) = 2*A204388(n). - Timothy L. Tiffin, Aug 09 2016

Extended by Vladimir Joseph Stephan Orlovsky, Feb 27 2011
a(29)-a(39) from Robert Price, Oct 15 2015
a(40)-a(48) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 25 2023

A220077 Numbers k such that 2^k + 35 is prime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 25, 33, 57, 117, 133, 189, 195, 263, 273, 287, 509, 693, 1087, 1145, 1159, 1845, 2743, 3275, 12223, 26263, 31425, 44359, 48003, 49251, 62557, 113877, 114507, 132865, 165789, 192549, 348437, 426043, 436365, 471043, 480417

Offset: 1

Vincenzo Librandi, Dec 04 2012

nonn

Some terms correspond to probable primes. Lifchitz link shows Lelio R Paula found the terms 132865, 165789, 192549, 348437. - Jens Kruse Andersen, Oct 01 2014
a(43) > 5*10^5. - Robert Price, Nov 01 2015
All terms are odd. - Elmo R. Oliveira, Nov 27 2023

Henri Lifchitz and Renaud Lifchitz, Search for 2^n+35, PRP Top Records.

Cf. A094076, A176927.

Cf. Numbers k such that 2^k + d is prime: (0,1,2,4,8,16) for d=1; A057732 (d=3), A059242 (d=5), A057195 (d=7), A057196 (d=9), A102633 (d=11), A102634 (d=13), A057197 (d=15), A057200 (d=17), A057221 (d=19), A057201 (d=21), A057203 (d=23), A157006 (d=25), A157007 (d=27), A156982 (d=29), A247952 (d=31), A247953 (d=33), this sequence (d=35).

Mathematica
```
Select[Range[5000],PrimeQ[2^# + 35] &]
```

for(n=1, 10^30, if (isprime(2^n + 35), print1(n", "))); \\ Altug Alkan, Oct 05 2015

a(26)-a(34) from Jens Kruse Andersen, Oct 01 2014
132865, 165789, 192549, 348437 discovered by Lelio R Paula confirmed as a(35)-a(38) by Robert Price, Oct 05 2015
a(39)-a(42) from Robert Price, Nov 01 2015

A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

Original entry on oeis.org

124, 9664, 151115727458150838697984

Offset: 1

Max Alekseyev, Jul 29 2025

Contains numbers 2^(k-1)*(2^k + 23) for k in A057203. First three terms have this form.

Deficiency k: A191363 (k=2), A125246 (k=4), A141548 (k=6), A125247 (k=8), A101223 (k=10), A141549 (k=12), A141550 (k=14), A125248 (k=16), A223608 (k=18), A223607 (k=20), A223606 (k=22), A275702 (k=26).
Abundance k: A088831 (k=2), A088832 (k=4), A087167 (k=6), A088833 (k=8), A223609 (k=10), A141545 (k=12), A141546 (k=14), A141547 (k=16), A223610 (k=18), A223611 (k=20), A223612 (k=22), A223613 (k=24), A275701 (k=26).
Cf. A057203.

A176922 Primes of the form 2^k + 23.

Original entry on oeis.org

31, 151, 549755813911, 604462909807314587353111

Offset: 1

Vincenzo Librandi, Apr 29 2010

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..8

Cf. A000040, A057203.

[a: n in [0..750] | IsPrime(a) where a is 2^n+23];

Select[Table[2^n + 23, {n, 0, 750}], PrimeQ] (* Vincenzo Librandi Jan 02 2014 *)

a(n) = 2^A057203(n) + 23. [R. J. Mathar, Jun 18 2010]

A175236 Primes p such that 2^p+23 is also prime.

Original entry on oeis.org

3, 7, 79, 359, 1031, 1039, 11311, 47599

Offset: 1

Vincenzo Librandi, Mar 09 2010

a(9) > 5*10^5. - Robert Price, Sep 18 2015

			For p=3, 2^3+23=31; p=7, 2^7+23=131; p=79, 2^79+23=604462909807314587353111

[p: p in PrimesUpTo(1000) | IsPrime(2^p+23) ];

Select[Prime[Range[1000]], PrimeQ[2^#+23]&] (*Robert Price, Sep 18 2015 *)

A057203 INTERSECT A000040. [R. J. Mathar, Jul 06 2010]

a(5)-a(6) from R. J. Mathar, Jul 06 2010

a(7)-a(8) from Robert Price, Sep 18 2015

A361744 A(n,k) is the least m such that there are k primes in the set {prime(n) + 2^i | 1 <= i <= m}, or -1 if no such number exists; square array A(n,k), n > 1, k >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 1, 3, 3, 2, 4, 5, 4, 1, 6, 11, 6, 3, 2, 7, 47, 8, 5, 4, 1, 12, 53, 10, 7, 8, 13, 2, 15, 141, 16, 9, 20, 21, 6, 3, 16, 143, 18, 15, 38, 33, 30, 7, 1, 18, 191, 20, 23, 64, 81, 162, 39, 3, 4, 28, 273, 28, 29, 80, 129, 654, 79, 5, 12, 2

Offset: 2

Jean-Marc Rebert, Mar 22 2023

			p = prime(2) = 3, m=1, u = {p + 2^k | 1 <= k <= m} = {5} contains one prime, and no lesser m satisfies this, so A(2,1) = 1.
Square array A(n,k) n > 1 and k >= 1 begins:
 1,     2,     3,     4,     6,     7,    12,    15,    16,    18, ...
 1,     3,     5,    11,    47,    53,   141,   143,   191,   273, ...
 2,     4,     6,     8,    10,    16,    18,    20,    28,    30, ...
 1,     3,     5,     7,     9,    15,    23,    29,    31,    55, ...
 2,     4,     8,    20,    38,    64,    80,   292,  1132,  4108, ...
 1,    13,    21,    33,    81,   129,   285,   297,   769,  3381, ...
 2,     6,    30,   162,   654,   714,  1370,  1662,  1722,  2810, ...
 3,     7,    39,    79,   359,   451,  1031,  1039, 11311, 30227, ...
 1,     3,     5,     7,     9,    13,    15,    17,    23,    27, ...

Cf. A057732 (1st row), A094076 (1st column).
Cf. A361679.
Cf. A019434 (primes 2^n+1), A057732 (2^n+3), A059242 (2^n+5), A057195 (2^n+7), A057196(2^n+9), A102633 (2^n+11), A102634 (2^n+13), A057197 (2^n+15), A057200 (2^n+17), A057221 (2^n+19), A057201 (2^n+21), A057203 (2^n+23).
Cf. A205558 and A231232 (with 2*k instead of 2^k).

A(n, k)= {my(nb=0, p=prime(n), m=1); while (nb

cp's OEIS Frontend

A247952 Numbers k such that 2^k + 31 is prime.

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Magma

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A247953 Numbers k such that 2^k + 33 is prime.

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Magma

Maple

Mathematica

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A156982 Numbers k such that 2^k + 29 is prime.

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Magma

Mathematica

PARI

Extensions

A157006 Numbers k such that 2^k + 25 is prime.

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Magma

Mathematica

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A220077 Numbers k such that 2^k + 35 is prime.

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Mathematica

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A385255 Numbers m whose deficiency is 24: sigma(m) - 2*m = -24.

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A176922 Primes of the form 2^k + 23.

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