A085398 -id:A085398 - cp's OEIS frontend

A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).

2, 4, 2, 6, 2, 20, 20, 26, 25, 10, 14, 5, 373, 4, 65, 232, 56, 2, 521, 911, 1156, 1619, 647, 511, 34, 2336, 2123, 1274, 2866, 951, 2199, 1353, 4965, 7396, 13513, 3692, 14103, 32275, 2257, 86, 3928, 2779, 18781, 85835, 820, 16647, 2468, 26677, 1172, 38361, 40842

Offset: 1

Lei Zhou, Apr 04 2012

1 - m^k + m^(2*k) - m^(3^k) + m^(4*k) equals Phi(10*k,m).
First 15 terms were generated by the provided Mathematica program. All other terms found using OpenPFGW as Fermat and Lucas PRP. Term 16-20, 22-24, 27 have N^2-1 factored over 33.3% and proved using OpenPFGW;
terms 21, 25, 29-33, 36, 37, 39, 41, 42, 45, 48, 51 are proved using CHG pari script;
terms 26, 28, 34, 40 are proved using kppm PARI script;
terms 35, 38, 43, 44, 46, 47, 49, 50 do not yet have a primality certificate.
The corresponding prime number of term 51 (40842) has 236089 digits.
The corresponding prime numbers for the following terms are equal:
  p(3)  = p(2)  = Phi(10, 2^4),
  p(12) = p(9)  = Phi(10, 5^50),
  p(18) = p(14) = Phi(10, 2^160),
  p(25) = p(21) = Phi(10, 34^512),
  p(40) = p(34) = Phi(10, 86^4000).

			n=1, A003592(1) = 1, when a=2, 1 - 2^1 + 2^2 - 2^3 + 2^4 = 11 is prime, so a(1)=2;
n=2, A003592(2) = 2, when a=4, 1 - 4^2 + 4^4 - 4^6 + 4^8 = 61681 is prime, so a(2)=4;
...
n=13, A003592(13) = 64, when a=373, PrimeQ(1 - 373^64 + 373^128 - 373^192 + 373^256) = True, while for a = 2..372, PrimeQ(1 - a^64 + a^128 - a^192 + a^256) = False, so a(13)=373.

Lei Zhou, Prime certificates of the corresponding primes of this sequence, April 2012.

Cf. A003592, A085398, A153438, A205506, A206418.

fQ[n_] := PowerMod[10, n, n] == 0;a = Select[10 Range@100, fQ]/10; l = Length[a]; Table[m = a[[j]]; i = 1;While[i++; cp = 1 - i^m + i^(2*m)-i^(3*m)+i^(4*m); ! PrimeQ[cp]]; i, {j, 1, l}]

do(k)=my(m=1);while(!ispseudoprime(polcyclo(10*k,m++)),);m
list(lim)=my(v=List(), N); for(n=0, log(lim)\log(5), N=5^n; while(N<=lim, listput(v, N); N<<=1)); apply(do, vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 04 2012

a(n) = A085398(10*A003592(n)). - Jinyuan Wang, Jan 01 2023

Term 50 added and comments updated by Lei Zhou, Jul 27 2012

Term 51 added and comments updated by Lei Zhou, Oct 10 2012

A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 7, 2, 11, 3, 2, 113, 2, 43, 2, 2, 5, 151, 2, 2, 2, 2, 2, 179, 3, 61, 2, 23, 2, 53, 2, 89, 137, 11, 2, 5, 5, 2, 7, 73, 11, 307, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 0, 2, 53, 491, 197, 2, 3, 3, 3, 11, 19, 59, 7, 2, 2, 271, 2, 191, 61, 41, 7, 2, 2, 59, 5, 2, 2

Offset: 1

Eric Chen, Dec 18 2014

nonn

Conjecture: a(n) > 0 for all n != 2^k (k>5).
Clearly, if n is a power of 2, and Phi_n(2) is not prime, then a(n) = 0.
Records: 3, 5, 7, 11, 113, 151, 179, 307, 491, 839, 1427, 2411, 5987, 6389, 8933, 11813, 18587, 31721, 40763, 46349, ..., . - Robert G. Wilson v, May 21 2017

Robert G. Wilson v, Table of n, a(n) for n = 1..1300 (first 1000 terms from Charles R Greathouse IV)

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

Do[n=1; p=Prime[n]; cp=Cyclotomic[k, p]; While[!PrimeQ[cp], n=n+1; p=Prime[n]; cp=Cyclotomic[k, p]]; Print[p], {k, 1, 300}]

a(n)=if(n>>valuation(n,2)==1 && n>32, if(ispseudoprime(2^(n/2)+1), 2, 0), my(P=polcyclo(n)); forprime(p=2,, if(ispseudoprime(subst(P,'x,p)), return(p)))) \\ Charles R Greathouse IV, Dec 18 2014

A298206 a(n) is the smallest b >= 2 such that b^(62^n) - b^(32^n) + 1 is prime.

Original entry on oeis.org

6, 3, 3, 6, 5, 106, 207, 569, 224, 736, 2854, 21234, 14837, 165394, 24743, 62721, 237804, 143332

Offset: 0

Serge Batalov, Jan 14 2018

a(13) = 165394 is a significant outlier from the generally expected trend, which can be conjectured to be 6*2^n*gamma, where gamma is the Euler-Mascheroni constant A001620. Additionally, the next b > a(13) such that b^(6*2^n) - b^(3*2^n) + 1 is prime is 165836, which is remarkably close to a(13). - Serge Batalov, Jan 24 2018

			2^12 - 2^6 + 1 = 4033 is composite and 3^12 - 3^6 + 1 = 530713 is prime, so a(1) = 3.

Phil Carmody, Prime Internet Eisenstein Search (ca. 2004-2005)
Mersenneforum, Prime Internet Eisenstein Search discussion
The Prime Pages, Generalized unique primes

Subsequence of A205506.

Cf. A001620, A085398, A153438, A246119.

for(n=0,9,for(b=2,1000,x=b^(3*2^n); if(isprime(x*(x-1)+1), print1(b,", "); break)))

a(n) = A085398(18*2^n). - Jinyuan Wang, Dec 21 2022

a(13) from Serge Batalov, Jan 24 2018

A353101 Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.

Original entry on oeis.org

2, 2, 22, 2, 43, 24, 315, 38, 54, 265, 605, 61, 697, 306, 1153, 370, 2, 10688, 3075, 2338, 1153, 3243, 130, 2301, 315, 200, 1155, 14739, 4591, 2230, 263, 6665, 250, 10520, 2228, 3699, 1126, 8925, 8732, 10556, 19860, 29121, 32804, 4666, 2313, 27398, 14280, 2013, 29022, 26131, 21430, 21996, 95774, 49363, 12648, 54308, 6737, 8745, 11121, 49627

Offset: 1

Jeppe Stig Nielsen, Apr 24 2022

nonn

The expression is the cyclotomic polynomial value Phi_{p^2}(b) where p=prime(n).
By definition, a(n) > 1. The occurrences of a(n)=2 correspond exactly to the terms of A156585.
Does a(n) tend to infinity (is liminf a(n) infinite)?
If it exists, a(27) > 857. - J.W.L. (Jan) Eerland, Dec 23 2022
a(65) = 1624. - Serge Batalov, Nov 17 2023

MersenneForum, Primes of the form (b^p^n-1)/(b^p^(n-1)-1) search thread

Cf. A001248, A066180, A085398, A156585.

Table[k=2;Monitor[Parallelize[While[True,If[PrimeQ[(k^(Prime[n]^2)-1)/(k^Prime[n]-1)],Break[]];k++];k],k],{n,1,10}] (* J.W.L. (Jan) Eerland, Dec 22 2022 *)

forprime(p=2,,for(b=2,+oo,if(ispseudoprime(polcyclo(p^2,b)),print1(b,", ");break())))

from sympy import isprime, prime
def a(n, startb=2):
    pn = prime(n); pn2 = pn**2; b = startb
    while not isprime((b**pn2-1)//(b**pn-1)): b += 1
    return b
print([a(n) for n in range(1, 10)]) # Michael S. Branicky, Jun 21 2022

a(n) = A085398(prime(n)^2) = A085398(A001248(n)).

a(25)-a(26) from J.W.L. (Jan) Eerland, Dec 23 2022
a(27) from Michael S. Branicky, Apr 04 2023
a(28)-a(33) from Martin Hopf, Nov 10 2023
a(34)-a(60) from Ryan Propper, Nov 17 2023

cp's OEIS Frontend

A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).

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A252503 Smallest prime p such that Phi_n(p) is also prime, where Phi is the cyclotomic polynomial, or 0 if no such p exists.

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A298206 a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.

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A353101 Least b > 1 such that (b^(prime(n)^2) - 1)/(b^prime(n) - 1) is prime.

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A298206 a(n) is the smallest b >= 2 such that b^(62^n) - b^(32^n) + 1 is prime.