A000068 -id:A000068 - cp's OEIS frontend

A056995 Numbers k such that k^256 + 1 is prime.

1, 278, 614, 892, 898, 1348, 1494, 1574, 1938, 2116, 2122, 2278, 2762, 3434, 4094, 4204, 4728, 5712, 5744, 6066, 6508, 6930, 7022, 7332, 8524, 8644, 8762, 8808, 9024, 9142, 9412, 10892, 12206, 13220, 13222, 13246, 13370, 13738, 14114, 14930

Offset: 1

Robert G. Wilson v, Sep 06 2000

nonn

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

Simon Plouffe, Table of n, a(n) for n = 1..10000 (1000 first terms from T. D. Noe)
Yves Gallot, Generalized Fermat Prime Search.
Simon Plouffe, 146309 terms

Cf. A005574, A000068, A006314, A006313, A006315, A006316, A056994, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Do[ k = 1; While[ PowerMod[ n, 256, 2*k*256 + 1 ] != 2*k*256 && k < 10^3, k++ ]; If[ k == 10^3 && PrimeQ[ n^256 + 1 ], Print[ n ] ], {n, 2, 15000, 2} ]

isA056995(n) = isprime(n^256+1) \\ Michael B. Porter, Apr 01 2010

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

Original entry on oeis.org

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

A037896 Primes of the form k^4 + 1.

Original entry on oeis.org

2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001, 723394817, 916636177, 1049760001, 1416468497

Offset: 1

Donald S. McDonald, Feb 27 2000

From Bernard Schott, Apr 22 2019: (Start)
These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a perfect biquadrate: A307690, so this sequence is a subsequence of A078164 and A307690.
If p prime = k^4 + 1, phi(p) = k^4.
The last three Fermat primes in A019434 {17, 257, 65537} belong to this sequence; with F_k = 2^(2^k) + 1 and for k = 2, 3, 4, phi(F_k) = (2^(2^(k-2)))^4. (End)

			6^4 + 1 = 1297 is prime.

T. D. Noe, Table of n, a(n) for n=1..1000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
M. Lal, Primes of the form n^4 + 1, Math. Comp. 21 (1967), pp. 245-247.

Cf. A000068, A002523, A019434, A078164, A307690.

Subsequence of A002496, A078164 and A039770.

[n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019

Select[Range[200]^4+1,PrimeQ] (* Harvey P. Dale, Jul 20 2015 *)

j=[]; for(n=1,200, if(isprime(n^4+1),j=concat(j,n^4+1))); j

list(lim)=my(v=List([2]),p); forstep(k=2,sqrtnint(lim\1-1,4),2, if(isprime(p=k^4+1), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 31 2022

[n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019

a(n) = A002523(A000068(n)). - Elmo R. Oliveira, Feb 21 2025

Corrected and extended by Jason Earls, Jul 19 2001

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

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

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A056995 Numbers k such that k^256 + 1 is prime.

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

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A037896 Primes of the form k^4 + 1.

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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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Extensions

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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