A096818 -id:A096818 - cp's OEIS frontend

A096821 Solutions to Mod[sigma(x),x]=12 of the form p*(p+13)/2 where p is a prime of form=2^j-13, of which j exponents are listed in A096818.

24, 304, 127744, 33501184, 8589082624, 10384593717069654320312270165377024, 822752278660603021077484591278411581166520461101278617407586304, 15177100720513508366558296147058741458142670970377727126573378340391656803557965824

Offset: 1

Labos Elemer, Jul 13 2004

nonn

The relevant A084306 includes additional solution like 21, while A076496 contains solutions of 6k form too.

			8th term is (2^136)*(-13+2^137) with 83 decimal digits; n=4: a[4]=(2^3)*(16-13)=24;

F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.

Cf. A084306, A096518, A000203, A076496.

A050414 Numbers k such that 2^k - 3 is prime.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 14114, 14400, 16460, 16680, 20757, 26350, 30041, 34452, 36552, 42689, 44629, 50474, 66422, 69337, 116926, 119324, 123297, 189110, 241004, 247165, 284133, 354946, 394034, 702194, 750740, 840797, 1126380, 1215889, 1347744, 1762004, 2086750

Offset: 1

Jud McCranie, Dec 22 1999

nonn

With 65 known primes corresponding to k < 1762005, these primes appear to be more common than Mersenne primes.  Of course at this time, the larger terms correspond only to probable primes. - Paul Bourdelais, Feb 04 2012
The numbers 2^k-3 and 2^k-1 are both primes for k = 3, 5, ? The lesser number 2^p-3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ... (see A283266). - Thomas Ordowski, Sep 18 2015
The terms a(43)-a(49) were found by Paul Underwood, a(50)-a(51) found by M. Frind and P. Underwood, a(52) found by Gary Barnes, a(53)-a(58) found by M. Frind and P. Underwood, and a(59)-a(66) found by Paul Bourdelais (see link Henri Lifchitz and Renaud Lifchitz). - Elmo R. Oliveira, Dec 02 2023

			k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.

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. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).
For prime terms see A283266.
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), A356826 (d=29).

Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]

for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014

More terms from Robert G. Wilson v, Sep 15 2000
More terms from Andrey V. Kulsha, Feb 11 2001
a(40) verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 2008
a(41) a new PRP term, from Serge Batalov, Oct 20 2008
Corrected and extended by including two smaller (apparently known) PRP and 16 larger terms from PRP Top Records of this form, all discovered by M. Frind & P. Underwood, Gary Barnes, Oct 20 2008
a(59)-a(60) discovered by Paul Bourdelais, Mar 26 2012
a(61)-a(63) discovered by Paul Bourdelais, Jun 18 2019
a(64) discovered by Paul Bourdelais, Jul 16 2019
a(65) discovered by Paul Bourdelais, Apr 20 2020
a(66) discovered by Paul Bourdelais, May 28 2020

A059608 Numbers k such that 2^k - 5 is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 18, 20, 26, 32, 36, 56, 66, 118, 130, 150, 166, 206, 226, 550, 706, 810, 1136, 1228, 1818, 2368, 2400, 3128, 4532, 5112, 8492, 16028, 16386, 17392, 18582, 21986, 24292, 27618, 30918, 32762, 48212, 120440, 183632, 316140, 364982, 414032, 533350, 595122

Offset: 1

Andrey V. Kulsha, Jan 30 2001

nonn

Except 3, all terms are even since for odd k, 2^k - 5 is divisible by 3.

			k = 10: 2^10 - 5 = 1019 is prime.
k = 20: 2^20 - 5 = 1048571 is prime.

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 (Editors), Search for 2^n-5, PRP Top Records.

Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), this sequence (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), A356826 (d=29).

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

is(n)=isprime(2^n-5) \\ Charles R Greathouse IV, Feb 17 2017

a(32)-a(34) from Labos Elemer, Jul 09 2004
a(35)-a(40) from Max Alekseyev, a(41) from Paul Underwood, a(42)-a(46) from Henri Lifchitz, added by Max Alekseyev, Feb 09 2012
a(47)-a(48) from Jon Grantham, Jul 29 2023

A059610 Numbers k such that 2^k - 9 is prime.

Original entry on oeis.org

4, 5, 9, 11, 17, 21, 33, 125, 141, 243, 251, 285, 321, 537, 563, 699, 729, 2841, 3365, 8451, 8577, 9699, 9725, 21011, 22689, 33921, 51761, 655845, 676761, 3480081

Offset: 1

Andrey V. Kulsha, Feb 02 2001

Except the first term 4, all terms are odd since 2^(2*m) - 9 = (2^m - 3)*(2^m + 3) is not prime for m > 2.

			243 is in the sequence because 2^243 - 9 is prime.

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. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), this sequence (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^#-9]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)

is(n)=isprime(2^n-9) \\ Charles R Greathouse IV, Feb 17 2017

a(24)-a(25) from Max Alekseyev, a(26)-a(27) from Paul Underwood, added by Max Alekseyev, Feb 09 2012
a(28)-a(29) from Robert Price, Jan 25 2017
a(30) 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

cp's OEIS Frontend

A096821 Solutions to Mod[sigma(x),x]=12 of the form p*(p+13)/2 where p is a prime of form=2^j-13, of which j exponents are listed in A096818.

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A050414 Numbers k such that 2^k - 3 is prime.

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A059608 Numbers k such that 2^k - 5 is prime.

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A059610 Numbers k such that 2^k - 9 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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Maple

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

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Sage