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Search: a059612 -id:a059612
Displaying 1-10 of 13 results found. page 1 2
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A050414 Numbers k such that 2^k - 3 is prime. +10
42
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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, ... - 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
LINKS
Table of n, a(n) for n=1..66.
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.
EXAMPLE
k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.
MATHEMATICA
Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]
PROG
(PARI) for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014
CROSSREFS
Cf. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).
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), A356826 (d=29).
KEYWORD
nonn
AUTHOR
Jud McCranie, Dec 22 1999
EXTENSIONS
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
STATUS
approved
A059608 Numbers k such that 2^k - 5 is prime. +10
25
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Except 3, all terms are even since for odd k, 2^k - 5 is divisible by 3.
LINKS
Table of n, a(n) for n=1..48.
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.
EXAMPLE
k = 10: 2^10 - 5 = 1019 is prime.
k = 20: 2^20 - 5 = 1048571 is prime.
MATHEMATICA
Select[Range[2, 20000], PrimeQ[2^# - 5] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
PROG
(PARI) is(n)=isprime(2^n-5) \\ Charles R Greathouse IV, Feb 17 2017
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
Andrey V. Kulsha, Jan 30 2001
EXTENSIONS
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
STATUS
approved
A096818 Numbers k such that 2^k - 13 is prime. +10
19
4, 5, 9, 13, 17, 57, 105, 137, 3217, 3229, 4233, 6097, 8757, 11457, 12073, 15425, 40117, 45357, 334809, 1509037 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Except the first term 4, all terms are odd since for even k, 2^k - 13 is divisible by 3.
LINKS
Table of n, a(n) for n=1..20.
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.
EXAMPLE
k = 5: 32 - 13 = 19 is prime.
MATHEMATICA
Select[Range[4, 20000], PrimeQ[2^#-13]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Jul 13 2004
EXTENSIONS
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
STATUS
approved
A059610 Numbers k such that 2^k - 9 is prime. +10
18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..30.
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.
EXAMPLE
243 is in the sequence because 2^243 - 9 is prime.
MATHEMATICA
Select[Range[3, 20000], PrimeQ[2^#-9]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
PROG
(PARI) is(n)=isprime(2^n-9) \\ Charles R Greathouse IV, Feb 17 2017
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Andrey V. Kulsha, Feb 02 2001
EXTENSIONS
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
STATUS
approved
A059609 Numbers k such that 2^k - 7 is prime. +10
17
39, 715, 1983, 2319, 2499, 3775, 12819, 63583, 121555, 121839, 468523, 908739 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
REFERENCES
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.
LINKS
Table of n, a(n) for n=1..12.
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.
EXAMPLE
k = 39, 2^39 - 7 = 549755813881 is prime.
MATHEMATICA
Select[Range[3, 20000], PrimeQ[2^# - 7] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011 *)
PROG
(PARI) is(n)=isprime(2^n-7) \\ Charles R Greathouse IV, Feb 17 2017
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Andrey V. Kulsha, Feb 02 2001
EXTENSIONS
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
STATUS
approved
A096820 Numbers k such that 2^k - 21 is prime. +10
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Similar to A057202 (which allows negative primes): this sequence is obtained by dropping the first four terms of A057202. - Joerg Arndt, Oct 05 2012
LINKS
Table of n, a(n) for n=1..38.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-21, PRP Top Records.
EXAMPLE
k = 5: 32 - 21 = 11 is prime.
k = 7: 128 - 21 = 107 is prime.
MATHEMATICA
Select[Range[5, 20000], PrimeQ[2^#-21]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)
PROG
(Sage)
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
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Jul 13 2004
EXTENSIONS
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
STATUS
approved
A059611 Numbers k such that 2^k - 17 is prime. +10
14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms are even since for odd k, 2^k - 17 is divisible by 3.
LINKS
Table of n, a(n) for n=1..48.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-17, PRP Top Records.
EXAMPLE
444 is present because 2^444 - 17 is prime.
MATHEMATICA
Select[Range[5, 20000], PrimeQ[2^#-17]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011 *)
PROG
(PARI) is(n)=isprime(2^n-17) \\ Charles R Greathouse IV, Feb 17 2017
CROSSREFS
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).
KEYWORD
nonn
AUTHOR
Andrey V. Kulsha, Feb 05 2001
EXTENSIONS
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
STATUS
approved
A096817 Numbers k such that 2^k - 11 is prime. +10
14
4, 6, 10, 18, 42, 78, 94, 114, 190, 322, 546, 3894, 10318, 11650, 12474, 20994, 61810, 103882, 296010, 636930, 653638, 926766 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
All terms are even since for odd k, 2^k - 11 is divisible by 3.
LINKS
Table of n, a(n) for n=1..22.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-11, PRP Top Records.
EXAMPLE
k = 6: 64 - 11 = 53 is prime.
MATHEMATICA
Select[Range[4, 20000], PrimeQ[2^#-11]&] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2011*)
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Jul 13 2004
EXTENSIONS
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
STATUS
approved
A096819 Numbers k such that 2^k - 19 is prime. +10
13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..29.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-19, PRP Top Records.
EXAMPLE
2^7 - 19 = 128 - 19 = 109, a prime, so 7 is a term of the sequence.
MATHEMATICA
Select[Range[5, 20000], PrimeQ[2^#-19]&] (* Vladimir Joseph Stephan Orlovsky, Feb 27 2011*)
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Labos Elemer, Jul 13 2004
EXTENSIONS
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
STATUS
approved
A057220 Numbers k such that 2^k - 23 is prime. +10
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..29.
Henri Lifchitz and Renaud Lifchitz, Search for 2^n-23, PRP Top Records.
EXAMPLE
k = 6: 2^6 - 23 = 41 is prime.
k = 8: 2^8 - 23 = 233 is prime.
MATHEMATICA
Do[ If[ PrimeQ[ 2^n - 23 ], Print[ n ] ], { n, 1, 15000} ]
PROG
(PARI) is(n)=ispseudoprime(2^n-23) \\ Charles R Greathouse IV, Jun 13 2017
CROSSREFS
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).
KEYWORD
nonn,more
AUTHOR
Robert G. Wilson v, Sep 16 2000
EXTENSIONS
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
STATUS
approved
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Last modified August 29 15:03 EDT 2024. Contains 375517 sequences. (Running on oeis4.)