A006315 -id:A006315 - cp's OEIS frontend

A057465 Numbers k such that k^512 + 1 is prime.

1, 46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, 4474, 4482, 4616, 4688, 5374, 5698, 5716, 5770, 6268, 6386, 6682, 7388, 7992, 8678, 8792, 9448, 9452, 9972, 10086, 10448, 10926, 11468, 12754, 13198, 13776, 14734, 16826, 16914, 18334

Offset: 1

Robert G. Wilson v, Sep 08 2000

nonn

Dubner, Harvey. "Generalized Fermat primes." J. Recreational Math., 18 (1985): 279-280.

T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search.

Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Do[ If[ PrimeQ[ n^512 + 1 ], Print[ n ] ], {n, 0, 31269} ]

isA057465(n) = isprime(n^512+1) \\ Michael B. Porter, Apr 02 2010

A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, 57394, 61686, 62060, 89762, 96632, 98242, 100540, 101578, 109696, 110540, 114690, 125440, 125442, 127596, 138068, 144362, 154908, 157310, 161822, 161900, 166224

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Yves Gallot, Generalized Fermat Prime Search.
Jeppe Stig Nielsen, All terms up to 2*10^9 (from a distributed search)
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
'stream', GFN-12 Final Statistic.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088362(n)=isprime(n^4096+1) \\ Michael B. Porter, Apr 22 2010

A088361 Numbers n such that n^2048 + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, 31614, 33308, 35054, 36702, 37062, 39020, 39056, 43738, 44174, 45654, 46502, 47348, 49190, 49204, 49544, 54514, 57210, 59770, 61184, 66894, 68194, 70574, 72446, 82642

Offset: 1

Jeppe Stig Nielsen, Sep 27 2003

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000 (from Yves Gallot)
Yves Gallot, Generalized Fermat Prime Search.
Jeppe Stig Nielsen, Generalized Fermat Primes sorted by base.
Eric Weisstein's World of Mathematics, Generalized Fermat Number.

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

isA088361(n)=isprime(n^2048+1) \\ Michael B. Porter, Apr 23 2010

A226528 Numbers k such that k^(2^13) + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894, 176206, 180924, 201170, 212724, 222764, 225174, 241600, 241860, 248744, 268032, 270674, 302368, 316970, 326260, 347962, 350830, 397468, 410938, 416010, 424584, 425848, 426338

Offset: 1

Robert G. Wilson v, Jun 09 2013

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000
PrimeGrid, GFN Prime Search Status and History: n=13.

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

is(n)=isprime(n^2^13+1) \\ Charles R Greathouse IV, Feb 17 2017

Missing terms inserted (from link) by Jeppe Stig Nielsen, Apr 14 2017

A226529 Numbers k such that k^(2^14) + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798, 422666, 426690, 449732, 462470, 468144, 498904, 506664, 509622, 528614, 572934, 581424, 638980, 641762, 656210, 698480, 704930, 730352, 795810, 840796, 908086, 975248, 976914, 990908, 1007874, 1037748, 1039970, 1067896, 1082054, 1097352, 1102754, 1132526, 1162996, 1171010, 1177808, 1181388

Offset: 1

Robert G. Wilson v, Jun 09 2013

Ray Chandler, Table of n, a(n) for n = 1..10000 (from PrimeGrid link below; n = 602..4825 from Jeppe Stig Nielsen)
PrimeGrid, GFN Prime Search Status and History: n=14.

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226530, A251597, A253854, A244150, A243959, A321323.

is(n)=isprime(n^2^14+1) \\ Charles R Greathouse IV, Feb 17 2017

A226530 Numbers b such that b^(2^15) + 1 is prime (a generalized Fermat prime).

Original entry on oeis.org

1, 70906, 167176, 204462, 249830, 321164, 330716, 332554, 429370, 499310, 524552, 553602, 743788, 825324, 831648, 855124, 999236, 1041870, 1074542, 1096382, 1113768, 1161054, 1167528, 1169486, 1171824, 1210354, 1217284, 1277444, 1519380, 1755378, 1909372, 1922592, 1986700, 2034902, 2147196, 2167350

Offset: 1

Robert G. Wilson v, Jun 09 2013

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..4346 (all terms below 460*10^6; terms 1..2729 from Ray Chandler)
PrimeGrid, GFN Prime Search Status and History: n=15.

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A251597, A253854, A244150, A243959, A321323.

is(b)=isprime(b^2^15+1) \\ Charles R Greathouse IV, Feb 17 2017

A244150 Numbers b such that b^262144+1 is prime.

Original entry on oeis.org

1, 24518, 40734, 145310, 361658, 525094, 676754, 773620, 1415198, 1488256, 1615588, 1828858, 2042774, 2514168, 2611294, 2676404, 3060772, 3547726, 3596074, 3673932, 3853792, 3933508, 4246258, 4489246, 5152128, 5205422, 5828034, 6287774, 6291332, 8521794

Offset: 1

Felix Fröhlich, Jun 21 2014

Base values b yielding a generalized Fermat prime b^(2^k)+1 for k=18.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..128
C. Caldwell, The largest known primes (Primes with 600,000 or more digits)
C. Caldwell, The Prime Pages Database Search Output
H. Dubner and Y. Gallot, Distribution of generalized Fermat prime numbers, Math. Comp., 71 (2002), 825-832.
M. Goetz, GFN Prime Discoveries, PrimeGrid forum.
J. S. S. Nielsen, Generalized Fermat Primes sorted by base (See table at the bottom of the page.)
PrimeGrid, Announcement of b=40734
PrimeGrid, Announcement of b=145310
PrimeGrid, Announcement of b=361658
PrimeGrid, Announcement of b=525094
PrimeGrid, Announcement of b=676754
PrimeGrid, Announcement of b=773620
PrimeGrid, Announcement of b=1415198
PrimeGrid, Announcement of b=1488256
PrimeGrid, Announcement of b=1615588
PrimeGrid, Announcement of b=1828858
PrimeGrid, Announcement of b=2042774
PrimeGrid, PrimeGrid Primes - (GFN262144) Generalized Fermat Prime Search n=262144
PrimePages, 1488256^262144 + 1
PrimePages, 1615588^262144 + 1

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A243959, A321323.

a(9), announced in message 92163 in PrimeGrid forum, added by Felix Fröhlich, Feb 17 2016
a(10), a(11) sent by Maximilian Pacher, Jun 27 2016, and a(12) on Aug 24 2016. - N. J. A. Sloane
a(13) from Felix Fröhlich, Nov 27 2016
a(14)-a(17) from Jeppe Stig Nielsen, Sep 06 2017
a(1) = 1 inserted by and more terms from Jeppe Stig Nielsen, Sep 10 2018
a(27)-a(30) from Jeppe Stig Nielsen, Sep 21 2019

A251597 Numbers b such that b^65536 + 1 is prime.

Original entry on oeis.org

1, 48594, 108368, 141146, 189590, 255694, 291726, 292550, 357868, 440846, 544118, 549868, 671600, 843832, 857678, 1024390, 1057476, 1087540, 1266062, 1361846, 1374038, 1478036, 1483076, 1540550, 1828502, 1874512, 1927034, 1966374, 2019300, 2041898, 2056292

Offset: 1

Felix Fröhlich, Dec 05 2014

nonn

Base values b yielding a generalized Fermat prime b^(2^k) + 1 for k=16.

First square member of sequence is 3934049284 = (A253854(1))^2. - Jeppe Stig Nielsen, Jun 29 2015

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..4048 (some previous contributions from Ray Chandler and Felix Fröhlich)
C. K. Caldwell, The Prime Database, Search Output.
Y. Gallot, Generalized Fermat Prime Search
Merfighters, Table of known GF primes b^n+1 where n(exponent) is at least 8192, Mersenne Wiki
J. S. S. Nielsen, Generalized Fermat Primes sorted by base (see table at the bottom of the page)
PrimeGrid, GFN Status by n-Range, Message 89145

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A253854, A244150, A243959, A321323.

Corrected last term, and extended, by Jeppe Stig Nielsen, Jun 29 2015
New b-file, updated with data from Message 89145 at PrimeGrid forum uploaded and sequence data corrected, by Felix Fröhlich, Jan 03 2016
a(1) = 1 inserted and new b-file by Jeppe Stig Nielsen, Sep 10 2018

A243959 Numbers k such that k^524288 + 1 is prime.

Original entry on oeis.org

1, 75898, 341112, 356926, 475856, 1880370, 2061748, 2312092, 2733014, 2788032, 2877652, 2985036, 3214654, 3638450, 4896418, 5897794, 6339004, 8630170, 9332124, 10913140, 11937916

Offset: 1

Felix Fröhlich, Jun 16 2014

Numbers k such that k^(2^j) + 1 is a generalized Fermat prime for j=19.
1880370 is a member, but its position is not yet known. - Jeppe Stig Nielsen, Jan 24 2018
PrimeGrid has now tested and double checked the necessary candidates to prove that 1880370 is a(6). - Jeppe Stig Nielsen, Feb 20 2018

C. Caldwell, The largest known primes (Primes with 600,000 or more digits)
H. Dubner and Y. Gallot, Distribution of generalized Fermat prime numbers, Math. Comp., 71 (2002), 825-832.
J. S. S. Nielsen, Generalized Fermat Primes sorted by base (See table at the bottom of the page.)
PrimeGrid, Announcement of n=75898
PrimeGrid, Announcement of n=341112
PrimeGrid, Announcement of n=356926
PrimeGrid, Announcement of n=475856
PrimeGrid, Announcement of n=1880370
PrimeGrid, Announcement of n=2061748
PrimeGrid, Announcement of n=2312092
PrimeGrid, GFN Prime Search Status and History
PrimeGrid, PrimeGrid Primes - (GFN524288) Generalized Fermat Prime Search n=524288
The Prime Pages, Database search results

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A321323.

is(n)=isprime(n^524288+1) \\ Charles R Greathouse IV, Feb 20 2017

a(6) from Jeppe Stig Nielsen, Feb 20 2018
a(7) from Jeppe Stig Nielsen, Apr 27 2018
a(1) = 1 inserted and a(8) added by Jeppe Stig Nielsen, Sep 10 2018
a(9)-a(12) from Jeppe Stig Nielsen, Sep 21 2019
a(13) from Jeppe Stig Nielsen, Dec 27 2019
a(14) from Ray Chandler, Mar 28 2022
a(15)-a(17) communicated by Jeppe Stig Nielsen, Apr 01 2024
a(18)-a(21) from Jeppe Stig Nielsen, Jan 11 2025

A253854 Numbers b such that b^131072 + 1 is prime.

Original entry on oeis.org

1, 62722, 130816, 228188, 386892, 572186, 689186, 909548, 1063730, 1176694, 1361244, 1372930, 1560730, 1660830, 1717162, 1722230, 1766192, 1955556, 2194180, 2280466, 2639850, 3450080, 3615210, 3814944, 4085818, 4329134, 4893072, 4974408, 5326454, 5400728, 5471814

Offset: 1

Felix Fröhlich, Jan 17 2015

Base values b yielding a generalized Fermat prime b^(2^k)+1 for k=17.

The first member exceeding 10^((10^6-1)/2^17) is known to be 42654182. - Jeppe Stig Nielsen, Jan 30 2016

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1243
C. K. Caldwell, 1560730^131072+1, The Largest Known Primes
C. K. Caldwell, Search result %^131072+1.
J. S. S. Nielsen, Generalized Fermat Primes sorted by base (see table at the bottom of the page)
PrimeGrid, PrimeGrid Primes - (GFN131072) Generalized Fermat Prime Search n=131072
PrimeGrid, Generalized Fermat statistics

Cf. A056993, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A244150, A243959, A321323.

Missing term a(8) inserted by Jeppe Stig Nielsen, Jul 02 2015
a(13) from Felix Fröhlich, Nov 01 2015
a(14)-a(20) from Jeppe Stig Nielsen, Jan 30 2016
a(21)-a(31) from Jeppe Stig Nielsen, Sep 06 2017
a(1) = 1 inserted by Jeppe Stig Nielsen, Sep 10 2018

cp's OEIS Frontend

A057465 Numbers k such that k^512 + 1 is prime.

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A088362 Numbers k such that k^4096 + 1 is prime (a generalized Fermat prime).

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A088361 Numbers n such that n^2048 + 1 is prime (a generalized Fermat prime).

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A226528 Numbers k such that k^(2^13) + 1 is prime (a generalized Fermat prime).

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A226529 Numbers k such that k^(2^14) + 1 is prime (a generalized Fermat prime).

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A226530 Numbers b such that b^(2^15) + 1 is prime (a generalized Fermat prime).

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A244150 Numbers b such that b^262144+1 is prime.

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A251597 Numbers b such that b^65536 + 1 is prime.

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A243959 Numbers k such that k^524288 + 1 is prime.

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A253854 Numbers b such that b^131072 + 1 is prime.

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