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

1,1

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, April 2012.

Formula

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

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, A085398, A153438, A205506, 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

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

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

Status

approved