A085398 -id:A085398 - cp's OEIS frontend

A250217 Numbers k such that A085398(k) sets a new record value.

1, 11, 18, 23, 25, 37, 68, 113, 115, 179, 229, 421, 545, 1129, 1451, 1481, 1847, 1987, 3099, 3329, 4391

Offset: 1

Eric Chen, Dec 24 2014

Record values at positions A085398(a(n)): {3, 5, 6, 10, 22, 61, 85, 86, 268, 304, 606, 808, ...}. - Michael De Vlieger, May 20 2017

Cf. A085398, A250218.

Function[s, Function[t, Map[Position[s, #][[1, 1]] &, t]]@ Union@ FoldList[Max, s]]@ Table[k = 2; While[! PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 500}] (* Michael De Vlieger, May 20 2017, after Eric Chen at A085398 *)

Corrected by insertion of 421. - Robert G. Wilson v, May 19 2017

a(17)-a(21) from Jinyuan Wang, Dec 17 2022

A250218 Record values in A085398.

Original entry on oeis.org

3, 5, 6, 10, 22, 61, 85, 86, 268, 304, 606, 808, 2706, 3771, 5047, 6288, 6298, 7631, 7643, 14683, 27620

Offset: 1

Eric Chen, Dec 24 2014

Cf. A085398, A250217.

Corrected by the insertion of 808. - Robert G. Wilson v, May 19 2017

a(17)-a(21) from Jinyuan Wang, Dec 17 2022

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 893 terms from Robert Price)
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (10), A162862 (11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060884, A085398, A259257.

[n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014

A246392:=n->`if`(isprime((n^5+1)/(n+1)),n,NULL): seq(A246392(n), n=1..500); # Wesley Ivan Hurt, Nov 15 2014

Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)

for(n=1,10^3,if(isprime(polcyclo(10,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

Original entry on oeis.org

2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, 919444

Offset: 0

Robert G. Wilson v, Sep 06 2000

Smallest base value yielding generalized Fermat primes. - Hugo Pfoertner, Jul 01 2003
The first 5 terms correspond with the known (ordinary) Fermat primes. A probable candidate for the next entry is 62722^131072+1, discovered by Michael Angel in 2003. It has 628808 decimal digits. - Hugo Pfoertner, Jul 01 2003
For any n, a(n+1) >= sqrt(a(n)), because k^(2^(n+1))+1 = (k^2)^(2^n)+1. - Jeppe Stig Nielsen, Sep 16 2015
Does the sequence contain any perfect squares? If a(n) is a perfect square, then a(n+1) = sqrt(a(n)). - Jeppe Stig Nielsen, Sep 16 2015
If for a particular n, a(n) exists, then a(i) exist for all i=0,1,2,...,n. No proof is known that this sequence is infinite. Such a result would clearly imply the infinitude of A002496. - Jeppe Stig Nielsen, Sep 18 2015
919444 is a candidate for a(20). See Zimmermann link. -  Serge Batalov, Sep 02 2017
Now PrimeGrid has tested and double checked all b^(2^20) + 1 with b < 919444, so we have proof that a(20) = 919444. - Jeppe Stig Nielsen, Dec 30 2017

			The primes are 2^(2^0) + 1 = 3, 2^(2^1) + 1 = 5, 2^(2^2) + 1 = 17, 2^(2^3) + 1 = 257, 2^(2^4) + 1 = 65537, 30^(2^5) + 1, 102^(2^6) + 1, ....

Yves Gallot, Generalized Fermat Prime Search
Lucile and Yves Gallot, Generalized Fermat Prime Search
Michael Goetz, id=103235 of Top 5000 Primes
Luke Harmon, Gaetan Delavignette, Arnab Roy, and David Silva, PIE: p-adic Encoding for High-Precision Arithmetic in Homomorphic Encryption, Cryptology ePrint Archive 2023/700.
Stephen Scott, id=84401 of Top 5000 Primes
Sylvanus A. Zimmerman, PrimeGrid’s Generalized Fermat Prime Search

Cf. A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323.

Cf. A019434 (Fermat primes).

f[n_] := (p = 2^n; k = 2; While[cp = k^p + 1; !PrimeQ@cp, k++ ]; k); Do[ Print[{n, f@n}], {n, 0, 17}] (* Lei Zhou, Feb 21 2005 *)

a(n)=my(k=2);while(!isprime(k^(2^n)+1),k++);k \\ Anders Hellström, Sep 16 2015

a(n) = A085398(2^(n+1)). - Jianing Song, Jun 13 2022

from Robert G. Wilson v, Oct 30 2000
from Jeppe Stig Nielsen, Aug 07 2005
and 75898 from Lei Zhou, Feb 01 2012
919444 from Jeppe Stig Nielsen, Dec 30 2017

A084740 Least k such that (n^k-1)/(n-1) is prime, or 0 if no such prime exists.

Original entry on oeis.org

2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, 0, 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2, 3, 2, 19, 97, 3, 2, 17, 2, 17, 3, 3, 2, 23, 29, 7, 59

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

When (n^k-1)/(n-1) is prime, k must be prime. As mentioned by Dubner, when n is a perfect power, then (n^k-1)/(n-1) will usually be composite for all k, which is the case for n = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, ... - T. D. Noe, Jan 30 2004
a(152) > prime(1100) or 0. - Derek Orr, Nov 29 2014
a(n)=2 if and only if n=p-1, where p is an odd prime; that is, n belongs to A006093, except 2. - Thomas Ordowski, Sep 19 2015
Probably a(152) = 270217 since (152^270217-1)/(152-1) has been shown to be probably prime. - Michael Stocker, Jan 24 2019

			a(7) = 5 as (7^5 - 1 )/(7 - 1) = 2801 = 1 + 7 + 7^2 + 7^3 + 7^4 is a prime but no smaller partial sum yields a prime.

Eric Chen, Table of known a(n) up to a(360) [with a(316) corrected by Jon E. Schoenfield]
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Andy Steward, Titanic Prime Generalized Repunits
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.
Robert G. Wilson v, Table of known a(n) from n = 2 to 1000.

Cf. A065507, A065854, A084738, A084742, A096059, A126659, A128164, A246005.

Cf. A066180, A085398, A103795, A123487, A123627.

a(n) = {l=List([9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343]); for(q=1, #l, if(n==l[q], return(0))); k=1; while(k, s=(n^prime(k)-1)/(n-1); if(ispseudoprime(s), return(prime(k))); k++)}
n=2; while(n<361, print1(a(n), ", "); n++) \\ Derek Orr, Jul 13 2014

More terms from T. D. Noe, Jan 23 2004

A066180 a(n) = smallest base b so that repunit (b^prime(n) - 1) / (b - 1) is prime, where prime(n) = n-th prime; or 0 if no such base exists.

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39

Offset: 1

Frank Ellermann, Dec 15 2001

nonn

Is a(n) = 0 possible?
Let p be the n-th prime; Cp(x) be the p-th cyclotomic polynomial (x^p - 1)/(x - 1); a(n) is the least k > 1 such that Cp(k) is prime.
The values associated with a(5) and a(8) through a(70) have been certified prime with Primo. (a(1) through a(4), a(6) and a(7) give prime(2), prime(4), prime(11), prime(31), prime(1028) and prime(12251), respectively.)

			a(5) = 5 because 11 is the 5th prime; (b^5 - 1)/(b - 1) is composite for b = 2,3,4 and prime ((5^11 - 1)/4 = 12207031) for b = 5.
b = 61 for prime(12) = 37 because (61^37 - 1)/60 is prime and 61 is the least base b that makes (b^37 - 1)/(b - 1) a prime.

Paulo Ribenboim, "The New Book of Prime Numbers Records", Springer, 1996, p. 353.

Robert G. Wilson v, Table of n, a(n) for n = 1..300 (terms 1..200 from Charles R Greathouse IV).
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Andy Steward, Titanic Prime Generalized Repunits.
Eric Weisstein's World of Mathematics, Repunit.
H. C. Williams and E. Seah, Some primes of the form: (a^n - 1)/(a - 1), Mathematics of Computation 23, 1979.

Cf. A004023 (prime repunits in base 10), A000043 (prime repunits in base 2, Mersenne primes), A055129 (table of repunits).

Cf. A084732, A085398.

Table[p = Prime[n]; b = 1; While[b++; ! PrimeQ[(b^p - 1)/(b - 1)]]; b, {n, 1, 70}] (* Lei Zhou, Oct 07 2011 *)

/* This program assumes (probable) primes exist for each n. */
/* All 70 (probable) primes found by this program have been proved prime. */
gen_repunit(b,n) = (b^prime(n)-1)/(b-1);
for(n=1,70, b=1; until(isprime(p), b++; p=gen_repunit(b,n)); print1(b,","));

a(n) = A085398(prime(n)).

Sequence extended to 16 terms by Don Reble, Dec 18 2001
More terms from Rick L. Shepherd, Sep 14 2002
Entry revised by N. J. A. Sloane, Jul 23 2006

A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

Original entry on oeis.org

2, 2, 2, 21, 209, 72, 260, 17, 3311, 4469, 94259, 55599

Offset: 0

Pierre CAMI, Dec 26 2008

Numbers of the form k^n*(k^n+1) + 1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even k^(4*n) + k^(2*n) + 1 = (k^(2*n)+1)^2 - (k^n)^2 = (k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3 is k^n*(k^n+1) + 1 always composite?
Phi[3^(n+1),k] = k^(3^n)*(k^(3^n)+1)+1. When m <> 3^n in k^m*(k^m+1)+1, Phi[3m,k] < k^m*(k^m+1)+1 and is a divisor of it. - Lei Zhou, Feb 09 2012
The prime number corresponding to the 10th term is a 587458-digit number. - Lei Zhou, Jul 04 2014
x^(2*k) + x^k + 1 = (x^(3*k) - 1)/(x^k - 1) is the product over n dividing 3k but not dividing k of cyclotomic polynomials Phi(n). If k is a power of 3, n = 3k is the only such divisor and we have a single irreducible cyclotomic polynomial Phi(3k). Otherwise we have the product of more than one polynomial, with integer values > 1 for integer x > 1, and thus always composite numbers. - Martin Becker, Jun 22 2021

Lei Zhou, Prime Database Entry, July 04, 2014.
R. Propper and S. Batalov, Prime Database Entry, Nov 17, 2023.

Cf. A056993, A085398, A101406, A153436.

Table[i = 1; m = 3^u; While[i++; cp = 1 + i^m + i^(2*m); ! PrimeQ[cp]]; i, {u, 1, 7}] (* Lei Zhou, Feb 01 2012 *)

a(n) = my(k=2); while (!isprime(k^(3^n)*(k^(3^n)+1) + 1), k++); k; \\ Michel Marcus, Jan 01 2023

a(n) = A085398(3^(n+1)). - Jinyuan Wang, Jan 01 2023

3311 from Lei Zhou using OpenPFGW, Feb 01 2012
4469 from Lei Zhou using OpenPFGW, Feb 09 2012
New term, 94259, from Lei Zhou using OpenPFGW, Jul 04 2014
Name and Comment corrected by Robert Price, Nov 11 2018
a(0) inserted by Jinyuan Wang, Jan 01 2023
a(11) from Ryan Propper and Serge Batalov, Nov 17 2023

A103795 Minimal base b such that (b^prime(n)+1)/(b+1) is prime.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2, 16, 23, 273, 2, 460, 22, 3, 36, 28, 329, 43, 69, 86

Offset: 2

Lei Zhou, Feb 23 2005

nonn

Conjecture: sequence is defined for any n>=2.

			(2^prime(2)+1)/(2+1) = 3 is prime, so a(2)=2;
(2^prime(10)+1)/(2+1) = 178956971 has a factor of 59;
(3^prime(10)+1)/(3+1) = 17157594341221 has a factor of 523;
...
(7^prime(10)+1)/(7+1) = 402488219476647465854701 is prime, so a(10)=7.

Cf. A056993, A066180, A085398.

Do[p=Prime[k]; n=2; cp=(n^p+1)/(n+1); While[ !PrimeQ[cp], n=n+1; cp=(n^p+1)/(n+1)]; Print[n], {k, 2, 200}]

a(n) = A085398(2*prime(n)) for n >= 2. - Jinyuan Wang, Dec 17 2022

A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

Original entry on oeis.org

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset: 1

Labos Elemer, May 02 2002

When n is prime, then the solutions are given in A088790.
No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019
All terms <= 1858 have been proven with PARI's implementation of ECPP.  All larger terms are BPSW PRPs.  There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Wikipedia, Aurifeuillean factorization

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862 (11), this sequence (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060886, A125258, A085398, A124990.

A246397:=n->`if`(isprime(n^4-n^2+1),n,NULL): seq(A246397(n),n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

for(n=1,10^3,if(isprime(polcyclo(12,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

cp's OEIS Frontend

A250217 Numbers k such that A085398(k) sets a new record value.

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A250218 Record values in A085398.

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A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

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A056993 a(n) is the smallest k >= 2 such that k^(2^n)+1 is prime, or -1 if no such k exists.

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A084740 Least k such that (n^k-1)/(n-1) is prime, or 0 if no such prime exists.

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A066180 a(n) = smallest base b so that repunit (b^prime(n) - 1) / (b - 1) is prime, where prime(n) = n-th prime; or 0 if no such base exists.

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A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

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A103795 Minimal base b such that (b^prime(n)+1)/(b+1) is prime.

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