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A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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].

LINKS

Table of n, a(n) for n=1..51.

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

EXAMPLE

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;

MATHEMATICA

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}]

PROG

(PARI) 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

CROSSREFS

Cf. A003592, A205506, A153438, A206418.

Sequence in context: A286601 A340071 A102128 * A230436 A105393 A182812

Adjacent sequences:  A181977 A181978 A181979 * A181981 A181982 A181983

KEYWORD

nonn,hard

AUTHOR

Lei Zhou, Apr 04 2012

EXTENSIONS

Added term 50 and updated comments - Lei Zhou, Jul 27 2012

Added term 51 and updated comments - Lei Zhou, Oct 10 2012

STATUS

approved

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Last modified January 17 04:29 EST 2021. Contains 340214 sequences. (Running on oeis4.)