A096502 -id:A096502 - cp's OEIS frontend

A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

3, 5, 3, 549755813881, 7, 5, 3, 17, 47, 13, 11, 41, 7, 5, 3, 97, 31, 29, 2011, 89, 23, 536870869, 19, 17, 79, 13, 11, 73, 7, 5, 3, 193, 191, 61, 59, 953, 439, 53, 179, 433, 47, 173, 43, 41, 167, 37, 163, 929, 31, 29, 67108763, 409, 23, 149, 19, 17, 911, 13, 11, 137

Offset: 1

Labos Elemer, Jul 13 2004

nonn

If 2n-1 is a provable Riesel number (A101036), then there exists a finite set of primes P(2n-1) such that every 2^x-(2n-1) > 0 is divisible by p(x) in P(2n-1). If some 2^x-(2n-1) = p(x), then a(n) = p(x). Otherwise, p(x) is a proper divisor of 2^x-(2n-1), which must be composite, and no a(n) exists.

For example, if n = 254602, then 2n-1 = 509203 is a provable Riesel number. Every 2^x-509203 > 0 is divisible by prime p(x) in P(509203) = {3,5,7,13,17,241}. 2^x-509203 > 0 implies x >= 19 implies 2^x-509203 > 241 >= p(x), so p(x) is a proper divisor and every 2^x-509203 is composite. Hence a(254602) does not exist.

			a(1) = 3 is the first Mersenne prime;
a(64) = 2^47 - 127 = 140737488355201, where 47 = A096502(64), 127 = 2*64 - 1.

T. D. Noe, Table of n, a(n) for n = 1..935

Cf. A096502.

f[n_]:=Module[{lst={},exp=Ceiling[Log[2,1+n]]},While[!PrimeQ[2^exp-n],exp++]; AppendTo[lst,2^exp-n]]; Flatten[f/@Range[1,1001,2]] (* Ivan N. Ianakiev, Mar 08 2016 *)

A096818 Numbers k such that 2^k - 13 is prime.

Original entry on oeis.org

4, 5, 9, 13, 17, 57, 105, 137, 3217, 3229, 4233, 6097, 8757, 11457, 12073, 15425, 40117, 45357, 334809, 1509037

Offset: 1

Labos Elemer, Jul 13 2004

Except the first term 4, all terms are odd since for even k, 2^k - 13 is divisible by 3.

			k = 5: 32 - 13 = 19 is prime.

F. Firoozbakht and M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-13, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), this sequence (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[4,20000],PrimeQ[2^#-13]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

a(16) from Max Alekseyev, a(17)-a(18) from Henri Lifchitz, a(19) from Lelio R Paula, added by Max Alekseyev, Feb 09 2012

a(20) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 17 2023

A059609 Numbers k such that 2^k - 7 is prime.

Original entry on oeis.org

39, 715, 1983, 2319, 2499, 3775, 12819, 63583, 121555, 121839, 468523, 908739

Offset: 1

Andrey V. Kulsha, Feb 02 2001

			k = 39, 2^39 - 7 = 549755813881 is prime.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 39, p. 15, Ellipses, Paris 2008.
J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 395 pp. 55; 218, Ellipses Paris 2004.
Wacław Sierpiński, Co wiemy, a czego nie wiemy o liczbach pierwszych. Warsaw: PZWS, 1961, pp. 46-47.
Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, pp. 31, 75.

Keith Conrad, Square patterns and infinitude of primes, University of Connecticut, 2019.
Jon Grantham and Andrew Granville, Fibonacci primes, primes of the form 2^n-k and beyond, arXiv:2307.07894 [math.NT], 2023.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-7, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), this sequence (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[3, 20000], PrimeQ[2^# - 7] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=isprime(2^n-7) \\ Charles R Greathouse IV, Feb 17 2017

a(8) from Henri Lifchitz, a(9)-a(10) from Gary Barnes, added by Max Alekseyev, Feb 09 2012
a(11) from Lelio R Paula, added by Max Alekseyev, Oct 25 2015
a(12) from Jon Grantham, Aug 09 2023

A059612 Numbers k such that 2^k - 15 is prime.

Original entry on oeis.org

5, 7, 8, 10, 14, 16, 23, 76, 95, 100, 158, 196, 235, 338, 620, 1646, 1850, 1891, 3833, 4394, 5194, 6017, 6070, 8824, 9955, 11399, 12250, 28723, 32057, 45494, 137359, 139627, 160654, 178819, 183284, 276391, 283466, 400571, 449030, 632815, 875518, 981016, 3511529

Offset: 1

Andrey V. Kulsha, Feb 13 2001

nonn

			100 is present because 2^100 - 15 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-15, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), this sequence (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

A059612:=n->if isprime(2^n-15) then n; fi; seq(A059612(n), n=1..20000); # Wesley Ivan Hurt, Dec 06 2013

Select[Range[4,20000],PrimeQ[2^#-15]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

is(n)=isprime(2^n-15) \\ Charles R Greathouse IV, Feb 17 2017

a(26) from Labos Elemer, Jul 09 2004
a(27)-a(29) from Max Alekseyev, a(30) from Henri Lifchitz, a(31)-a(32) from Gary Barnes, a(33)-a(35) from Lelio R Paula, added by Max Alekseyev, Feb 09 2012
a(36)-a(37) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(38) from Lelio R Paula, added by Robert Price, Dec 06 2013
a(39) from Lelio R Paula, added by Robert Price, Mar 16 2019
a(40)-a(43) from Stefano Morozzi, added by Elmo R. Oliveira, Nov 16 2023

A096820 Numbers k such that 2^k - 21 is prime.

Original entry on oeis.org

5, 6, 7, 9, 11, 13, 14, 21, 23, 41, 46, 89, 110, 389, 413, 489, 869, 1589, 1713, 2831, 10843, 11257, 16949, 24513, 39621, 43349, 62941, 96094, 139237, 145289, 264683, 396790, 420694, 439931, 659589, 783893, 840203, 944561

Offset: 1

Labos Elemer, Jul 13 2004

Similar to A057202 (which allows negative primes): this sequence is obtained by dropping the first four terms of A057202. - Joerg Arndt, Oct 05 2012

			k = 5: 32 - 21 = 11 is prime.
k = 7: 128 - 21 = 107 is prime.

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

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), this sequence (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[5,20000],PrimeQ[2^#-21]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

def is_A096820(n):
    t = 2^n-21
    return t > 1 and is_prime(t)
def A096820_list(up_to):
    return [n for n in range(up_to) if is_A096820(n)]
A096820_list(100)  # Peter Luschny, Oct 04 2012

a(23)-a(24) from Max Alekseyev, a(25) from Donovan Johnson, a(26)-a(28) from Henri Lifchitz, a(29)-a(30) from Lelio R Paula, added by Max Alekseyev, Feb 10 2012
a(31)-a(32) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(33)-a(34) found by Lelio R Paula, a(35)-a(38) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 24 2023

A059611 Numbers k such that 2^k - 17 is prime.

Original entry on oeis.org

6, 8, 12, 16, 18, 20, 22, 24, 32, 36, 42, 44, 96, 104, 152, 174, 198, 336, 414, 444, 468, 488, 664, 808, 848, 3632, 4062, 5586, 5904, 6348, 8628, 9224, 9916, 13136, 15966, 17120, 17568, 17652, 20560, 31572, 33644, 104098, 115842, 130572, 164110, 189414, 205110, 406758

Offset: 1

Andrey V. Kulsha, Feb 05 2001

nonn

All terms are even since for odd k, 2^k - 17 is divisible by 3.

			444 is present because 2^444 - 17 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-17, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), this sequence (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[5,20000],PrimeQ[2^#-17]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)

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

a(34)-a(40) from Max Alekseyev, a(41) from Paul Underwood, a(42)-a(44) from Gary Barnes, a(45)-a(47) from Lelio R Paula, added by Max Alekseyev, Feb 09 2012

a(48) by Lelio R. Paula, added by Robert Price, Dec 06 2013

A096817 Numbers k such that 2^k - 11 is prime.

Original entry on oeis.org

4, 6, 10, 18, 42, 78, 94, 114, 190, 322, 546, 3894, 10318, 11650, 12474, 20994, 61810, 103882, 296010, 636930, 653638, 926766

Offset: 1

Labos Elemer, Jul 13 2004

All terms are even since for odd k, 2^k - 11 is divisible by 3.

			k = 6: 64 - 11 = 53 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-11, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), this sequence (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[4,20000],PrimeQ[2^#-11]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)

a(13)-a(16) from Max Alekseyev, a(17)-a(18) from Henri Lifchitz, added by Max Alekseyev, Feb 09 2012
a(19) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(20)-a(22) from Stefano Morozzi, added by Elmo R. Oliveira, Nov 16 2023

A096819 Numbers k such that 2^k - 19 is prime.

Original entry on oeis.org

5, 7, 11, 15, 19, 21, 31, 39, 67, 69, 85, 157, 171, 191, 255, 291, 379, 3669, 4551, 9531, 13119, 14211, 20647, 233965, 337267, 534429, 535415, 816039, 991715

Offset: 1

Labos Elemer, Jul 13 2004

All terms are odd since for even k, 2^k - 19 is divisible by 3.

a(26) > 5*10^5. - Tyler NeSmith, Apr 16 2022

			2^7 - 19 = 128 - 19 = 109, a prime, so 7 is a term of the sequence.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-19, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), this sequence (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).

Select[Range[5,20000],PrimeQ[2^#-19]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011*)

a(22)-a(23) from Max Alekseyev, Feb 10 2012
a(24)-a(25) from Lelio R Paula, added by Max Alekseyev, Oct 24 2013
a(26)-a(29) found by Stefano Morozzi, added by Alois P. Heinz, Aug 29 2022

A057220 Numbers k such that 2^k - 23 is prime.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 18, 36, 68, 152, 212, 324, 1434, 1592, 1668, 3338, 7908, 9662, 27968, 28116, 33974, 41774, 66804, 144518, 162954, 241032, 366218, 676592, 991968

Offset: 1

Robert G. Wilson v, Sep 16 2000

Note that for the values 2 and 4 the primes are negative.
a(22) > 41358. - Jinyuan Wang, Jan 20 2020
All terms are even. - Elmo R. Oliveira, Nov 24 2023

			k = 6: 2^6 - 23 = 41 is prime.
k = 8: 2^8 - 23 = 233 is prime.

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

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), this sequence (d=23), A356826 (d=29).

Do[ If[ PrimeQ[ 2^n - 23 ], Print[ n ] ], { n, 1, 15000} ]

is(n)=ispseudoprime(2^n-23) \\ Charles R Greathouse IV, Jun 13 2017

a(19)-a(21) from Jinyuan Wang, Jan 20 2020

a(22)-a(23) found by Henri Lifchitz, a(24)-a(27) found by Lelio R Paula, a(28)-a(29) found by Stefano Morozzi, added by Elmo R. Oliveira, Nov 24 2023

A356826 Numbers k such that 2^k - 29 is prime.

Original entry on oeis.org

5, 8, 104, 212, 79316, 102272, 225536, 340688

Offset: 1

Craig J. Beisel, Aug 29 2022

A particularly low-density pseudo-Mersenne sequence. I have verified that there are no additional terms for k < 5*10^4. For k = a(5), a(6), a(7), and a(8), 2^k - 29 is a probable prime (see link).
The terms a(5)-a(8) were discovered by Henri Lifchitz (see link). - Elmo R. Oliveira, Nov 29 2023
Empirically: except for 5, all terms are even. - Elmo R. Oliveira, Nov 29 2023

			5 is a term because 2^5 - 29 = 3 is prime.
8 is a term because 2^8 - 29 = 227 is prime.

Henri Lifchitz and Renaud Lifchitz, Search for 2^n-29, PRP Top Records.

Cf. A096502.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), this sequence (d=29).

for(n=2, 1000, if(isprime(2^n-29), print1(n, ", ")))

cp's OEIS Frontend

A096822 Smallest primes of form p = 2^x-(2n-1) where x=A096502(n), the least exponent providing this kind of prime.

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A096818 Numbers k such that 2^k - 13 is prime.

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A059609 Numbers k such that 2^k - 7 is prime.

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A059612 Numbers k such that 2^k - 15 is prime.

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A096820 Numbers k such that 2^k - 21 is prime.

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A059611 Numbers k such that 2^k - 17 is prime.

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A096817 Numbers k such that 2^k - 11 is prime.

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