A059242 -id:A059242 - cp's OEIS frontend

A057221 Numbers k such that 2^k + 19 is prime.

2, 6, 30, 162, 654, 714, 1370, 1662, 1722, 2810, 77142, 156254, 432974, 1092242, 1245230

Offset: 1

Robert G. Wilson v, Sep 17 2000

a(14) > 5*10^5. - Robert Price, Aug 27 2015
All terms are even. - Robert Israel, Aug 28 2015
For numbers k in this sequence, the number 2^(k-1)*(2^k+19) has deficiency 20 (see A223607). - M. F. Hasler, Jul 18 2016

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+19, PRP Top Records.

Cf. A223607, A253774.

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

[n: n in [0..1000] | IsPrime(2^n+19)]; // Vincenzo Librandi, Aug 28 2015

select(n -> isprime(2^n+19), [seq(2*i,i=1..10000)]); # Robert Israel, Aug 28 2015

Do[ If[ PrimeQ[ 2^n + 19 ], Print[ n ] ], {n, 1, 15000} ]
Select[Range[10000], PrimeQ[2^# + 19] &] (* Vincenzo Librandi, Aug 28 2015 *)

for(n=1,oo,ispseudoprime(2^n+19)&&print1(n",")) \\ M. F. Hasler, Jul 18 2016

a(n) = 2*A253774(n). - Joerg Arndt, Aug 28 2015

a(11)-a(13) from Robert Price, Aug 27 2015
Edited by M. F. Hasler, Jul 18 2016
a(14)-a(15) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 19 2023

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

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

Original entry on oeis.org

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

A217355 Numbers k such that 8^k + 5 is prime.

Original entry on oeis.org

1, 47, 91, 64655

Offset: 1

Vincenzo Librandi, Oct 02 2012

All terms are 1/3 of the terms of A059242 that are multiples of 3.

No more terms <= 10^5. - Tyler NeSmith

Cf. A059242, A217354.

[n: n in [0..1000] | IsPrime(8^n + 5)];

Mathematica
```
Select[Range[1000], PrimeQ[8^# + 5] &]
```

is(n)=ispseudoprime(8^n+5) \\ Charles R Greathouse IV, Feb 20 2017

a(4) from Tyler NeSmith, Jan 19 2021

A234285 Positive odd numbers n such that sigma(m) - 2m is never equal to n, where sigma(.) is the sum of divisors function A000203. Conjectural.

Original entry on oeis.org

1, 5, 9, 11, 13, 15, 21, 23, 25, 27, 29, 33, 35, 37, 43, 45

Offset: 1

N. J. A. Sloane, Dec 28 2013

The terms shown here are conjectural, based on a search up to 10^20 made by Davis et al. (2013).
Comments from Farideh Firoozbakht, Jan 12 2014: (Start)
1. Mersenne primes are not in this sequence. Because if M=2^p-1 is prime then M=sigma(m)-2m, where m=2^(p-1)*(2^p-1)^2=(1/2)*(M+1)*M^2 (please see Proposition 2.1 of Firoozbakht-Hasler, 2010).
2. If M = 2^p - 1 is a Mersenne prime then M^2 + 3M + 1 = 4^p + 2^p - 1 is not in the sequence. Because M^2 + 3M + 1 = sigma(m) - 2m where m = M^3 + M^2 = 2^p(2^p-1)^2 (please see Proposition 2.5, op. cit.).
Examples:
p = 2, M = 3, 4^p + 2^p - 1 = 19, m = M^3 + M^2 = 2^p(2^p-1)^2 = 36; sigma(m) - 2m = 19
p = 3, M = 7, 4^p + 2^p - 1 = 71, m = M^3 + M^2 = 2^p(2^p-1)^2 = 392; sigma(m) - 2m = 71
3. Note that if r is an even number and if for a number k p = 2^k - r - 1 is an odd prime then r = sigma(m) - 2m where m = 2^(k-1)*p. Namely r is not in the sequence (see Theorem 1.1, op. cit.).
It seems that for each even number r, there exists at least one odd prime of the form 2^k - r - 1. This means there is no even term in the sequence.
Moreover I conjecture that, for each even number r, there exist infinitely many primes p of the form 2^k - r - 1, or equivalently, I conjecture that: For each odd number s, there exists infinitely many primes p of the form 2^k - s.
Special cases:
(i): s = 1, there exist infinitely many Mersenne primes.
(ii): s = -1, there exist infinitely many Fermat primes.
(iii): s = 3, sequence A050414 is infinite.
(iv): s = -3, sequence A057732 is infinite.
(v): s = -5, sequence A059242 is infinite.
and so on. (End)
Cohen (1983) showed that 203^2 is not a term since sigma(m) - 2*m = 203^2 has a solution m = 742^2. - Max Alekseyev, Aug 29 2025

Graeme L. Cohen, Generalised quasiperfect numbers, Ph.D. Dissertation, University of New South Wales, Sydney, 1982. Abstract in Bull. Australian Math. Soc., 27 (1983), 153-155.
Nichole Davis, Dominic Klyve and Nicole Kraght, On the difference between an integer and the sum of its proper divisors, Involve, Vol. 6 (2013), No. 4, 493-504; DOI: 10.2140/involve.2013.6.493
Farideh Firoozbakht and M. F. Hasler, Variations on Euclid's formula for perfect numbers, Journal of Integer Sequences 13 (2010), 18 pp. #10.3.1.

Cf. A000203, A033879 (2n - sigma(n)).

For negative values of n see A234286.

Edited by Max Alekseyev, Aug 29 2025

A133830 Least positive number k < n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 0, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 0, 17, 15, 1, 15, 1, 6, 0, 4, 9, 14, 13, 3, 11, 25, 0, 6, 7, 0, 17, 7, 15, 2, 0, 30, 23, 6, 21, 2, 33, 1, 0, 3, 0, 14, 5, 6, 21, 19, 0, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27, 33, 4, 3, 26, 1, 39, 35, 19, 9, 18

Offset: 2

T. D. Noe, Sep 26 2007

nonn

Sequence A081504 gives the n such that a(n) = 0. For those n, A133831(n) gives the least k > n for which the binary trinomial is prime.

T. D. Noe, Table of n, a(n) for n=2..2000

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Cf. A095056, A133831, A133832 (k > n equivalent).

Mathematica
```
Table[s=1+2^n; k=1; While[k
				
```

Edited by Peter Munn, Sep 30 2024

cp's OEIS Frontend

A057221 Numbers k such that 2^k + 19 is prime.

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

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

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

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Extensions

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

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Magma

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Extensions

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

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Magma

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