Search: a059612 -id:a059612
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A050414
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Numbers k such that 2^k - 3 is prime.
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+10
42
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-3, PRP Top Records.
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EXAMPLE
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k = 22, 2^22 - 3 = 4194301 is prime.
k = 24, 2^24 - 3 = 16777213 is prime.
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MATHEMATICA
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Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]
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PROG
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(PARI) for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(40) verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 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
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STATUS
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approved
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A059608
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Numbers k such that 2^k - 5 is prime.
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+10
25
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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
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OFFSET
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1,1
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COMMENTS
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Except 3, all terms are even since for odd k, 2^k - 5 is divisible by 3.
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LINKS
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Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-5, PRP Top Records.
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EXAMPLE
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k = 10: 2^10 - 5 = 1019 is prime.
k = 20: 2^20 - 5 = 1048571 is prime.
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MATHEMATICA
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PROG
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A096818
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Numbers k such that 2^k - 13 is prime.
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+10
19
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4, 5, 9, 13, 17, 57, 105, 137, 3217, 3229, 4233, 6097, 8757, 11457, 12073, 15425, 40117, 45357, 334809, 1509037
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OFFSET
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1,1
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COMMENTS
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Except the first term 4, all terms are odd since for even k, 2^k - 13 is divisible by 3.
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LINKS
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EXAMPLE
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k = 5: 32 - 13 = 19 is prime.
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MATHEMATICA
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A059610
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Numbers k such that 2^k - 9 is prime.
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+10
18
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n-9, PRP Top Records.
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EXAMPLE
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243 is in the sequence because 2^243 - 9 is prime.
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MATHEMATICA
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PROG
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(24)-a(25) from Max Alekseyev, a(26)-a(27) from Paul Underwood, added by Max Alekseyev, Feb 09 2012
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STATUS
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approved
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A059609
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Numbers k such that 2^k - 7 is prime.
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+10
17
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39, 715, 1983, 2319, 2499, 3775, 12819, 63583, 121555, 121839, 468523, 908739
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OFFSET
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1,1
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REFERENCES
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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.
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LINKS
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EXAMPLE
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k = 39, 2^39 - 7 = 549755813881 is prime.
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MATHEMATICA
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PROG
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A096820
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Numbers k such that 2^k - 21 is prime.
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+10
15
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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
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OFFSET
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1,1
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COMMENTS
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Similar to A057202 (which allows negative primes): this sequence is obtained by dropping the first four terms of A057202. - Joerg Arndt, Oct 05 2012
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LINKS
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EXAMPLE
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k = 5: 32 - 21 = 11 is prime.
k = 7: 128 - 21 = 107 is prime.
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MATHEMATICA
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PROG
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(Sage)
t = 2^n-21
return t > 1 and is_prime(t)
return [n for n in range(up_to) if is_A096820(n)]
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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A059611
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Numbers k such that 2^k - 17 is prime.
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+10
14
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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
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OFFSET
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1,1
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COMMENTS
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All terms are even since for odd k, 2^k - 17 is divisible by 3.
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LINKS
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EXAMPLE
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444 is present because 2^444 - 17 is prime.
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MATHEMATICA
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PROG
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CROSSREFS
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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).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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A096817
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Numbers k such that 2^k - 11 is prime.
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+10
14
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4, 6, 10, 18, 42, 78, 94, 114, 190, 322, 546, 3894, 10318, 11650, 12474, 20994, 61810, 103882, 296010, 636930, 653638, 926766
(list;
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OFFSET
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1,1
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COMMENTS
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All terms are even since for odd k, 2^k - 11 is divisible by 3.
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LINKS
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EXAMPLE
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k = 6: 64 - 11 = 53 is prime.
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MATHEMATICA
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A096819
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Numbers k such that 2^k - 19 is prime.
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+10
13
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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;
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OFFSET
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1,1
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COMMENTS
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All terms are odd since for even k, 2^k - 19 is divisible by 3.
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LINKS
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EXAMPLE
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2^7 - 19 = 128 - 19 = 109, a prime, so 7 is a term of the sequence.
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MATHEMATICA
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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A057220
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Numbers k such that 2^k - 23 is prime.
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+10
11
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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
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OFFSET
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1,1
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COMMENTS
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Note that for the values 2 and 4 the primes are negative.
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LINKS
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EXAMPLE
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k = 6: 2^6 - 23 = 41 is prime.
k = 8: 2^8 - 23 = 233 is prime.
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MATHEMATICA
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Do[ If[ PrimeQ[ 2^n - 23 ], Print[ n ] ], { n, 1, 15000} ]
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PROG
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CROSSREFS
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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).
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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