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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).

%I A181980 #43 Nov 16 2023 07:43:27
%S A181980 2,4,2,6,2,20,20,26,25,10,14,5,373,4,65,232,56,2,521,911,1156,1619,
%T A181980 647,511,34,2336,2123,1274,2866,951,2199,1353,4965,7396,13513,3692,
%U A181980 14103,32275,2257,86,3928,2779,18781,85835,820,16647,2468,26677,1172,38361,40842
%N A181980 Least positive integer m > 1 such that 1 - m^k + m^(2k) - m^(3k) + m^(4k) is prime, where k = A003592(n).
%C A181980 1 - m^k + m^(2*k) - m^(3^k) + m^(4*k) equals Phi(10*k,m).
%C A181980 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;
%C A181980 terms 21, 25, 29-33, 36, 37, 39, 41, 42, 45, 48, 51 are proved using CHG pari script;
%C A181980 terms 26, 28, 34, 40 are proved using kppm PARI script;
%C A181980 terms 35, 38, 43, 44, 46, 47, 49, 50 do not yet have a primality certificate.
%C A181980 The corresponding prime number of term 51 (40842) has 236089 digits.
%C A181980 The corresponding prime numbers for the following terms are equal:
%C A181980   p(3)  = p(2)  = Phi(10, 2^4),
%C A181980   p(12) = p(9)  = Phi(10, 5^50),
%C A181980   p(18) = p(14) = Phi(10, 2^160),
%C A181980   p(25) = p(21) = Phi(10, 34^512),
%C A181980   p(40) = p(34) = Phi(10, 86^4000).
%H A181980 Lei Zhou, <a href="http://csic.som.emory.edu/~lzhou/blogs/?p=263">Prime certificates of the corresponding primes of this sequence</a>, April 2012.
%F A181980 a(n) = A085398(10*A003592(n)). - _Jinyuan Wang_, Jan 01 2023
%e A181980 n=1, A003592(1) = 1, when a=2, 1 - 2^1 + 2^2 - 2^3 + 2^4 = 11 is prime, so a(1)=2;
%e A181980 n=2, A003592(2) = 2, when a=4, 1 - 4^2 + 4^4 - 4^6 + 4^8 = 61681 is prime, so a(2)=4;
%e A181980 ...
%e A181980 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.
%t A181980 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}]
%o A181980 (PARI) do(k)=my(m=1);while(!ispseudoprime(polcyclo(10*k,m++)),);m
%o A181980 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
%Y A181980 Cf. A003592, A085398, A153438, A205506, A206418.
%K A181980 nonn,hard
%O A181980 1,1
%A A181980 _Lei Zhou_, Apr 04 2012
%E A181980 Term 50 added and comments updated by _Lei Zhou_, Jul 27 2012
%E A181980 Term 51 added and comments updated by _Lei Zhou_, Oct 10 2012