A005574 -id:A005574 - cp's OEIS frontend

A007591 Numbers k such that k^2 + 4 is prime.

1, 3, 5, 7, 13, 15, 17, 27, 33, 35, 37, 45, 47, 57, 65, 67, 73, 85, 87, 95, 97, 103, 115, 117, 125, 135, 137, 147, 155, 163, 167, 177, 183, 193, 203, 207, 215, 217, 233, 235, 243, 245, 253, 255, 265, 267, 275, 277, 287, 293, 303, 307, 313, 317, 347, 357, 373, 375

Offset: 1

N. J. A. Sloane, R. K. Guy

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Near-Square Prime

Other sequences of the type "Numbers k such that k^2 + i is prime": A005574 (i=1), A067201 (i=2), A049422 (i=3), this sequence (i=4), A078402 (i=5), A114269 (i=6), A114270 (i=7), A114271 (i=8), A114272 (i=9), A114273 (i=10), A114274 (i=11), A114275 (i=12).

[n: n in [0..400] | IsPrime(n^2+4)]; // Vincenzo Librandi, Nov 18 2010

lst={};Do[If[PrimeQ[n^2+4], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)
Select[Range[0, 400], PrimeQ[#^2 + 4] &] (* Vincenzo Librandi, Sep 25 2012 *)

select(n->isprime(n^2+4),vector(500,n,2*n-1)) \\ Charles R Greathouse IV, Sep 25 2012

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

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

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

A067200 Numbers m such that m^3 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 29, 45, 63, 65, 69, 71, 83, 105, 113, 123, 129, 143, 153, 171, 173, 189, 209, 215, 219, 231, 243, 245, 249, 263, 291, 299, 305, 311, 341, 363, 369, 395, 419, 425, 431, 435, 473, 483, 491, 495, 501, 503, 509, 515, 533, 549, 555, 561, 575, 579, 639

Offset: 1

Benoit Cloitre, Feb 19 2002

nonn

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A144953.

Other sequences of the type "Numbers m such that m^k + k - 1 is prime": A000040 (k=1), A005574 (k=2), this sequence (k=3), A125259 (k=4), A125260 (k=5), A125261 (k=6), A125262 (k=7), A125263 (k=8), A125264 (k=10), A125265 (k=11).

lst={}; Do[If[PrimeQ[n^3+2], AppendTo[lst, n]], {n, 0, 10^3}]; lst (* Vladimir Joseph Stephan Orlovsky, Sep 08 2008 *)
Select[Range[0,700],PrimeQ[#^3+2]&] (* Harvey P. Dale, Jul 27 2025 *)

is(n)=isprime(n^3 + 2) \\ Charles R Greathouse IV, Apr 29 2015

a(n) = (A144953(n) - 2)^(1/3). - Zak Seidov, Sep 16 2013

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

A007591 Numbers k such that k^2 + 4 is 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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A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is 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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A067200 Numbers m such that m^3 + 2 is prime.

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

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