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
Lei Zhou, Prime certificates of the corresponding primes of this sequence, April 2012.
FORMULA
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
KEYWORD
nonn,hard
AUTHOR
Lei Zhou, Apr 04 2012
EXTENSIONS
STATUS
approved