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A178070 Primes dividing repunits R(10^n) for some n.

%I A178070 #34 Nov 05 2024 19:04:55
%S A178070 11,17,41,73,101,137,251,257,271,353,401,449,641,751,1201,1409,1601,
%T A178070 3541,4001,4801,5051,9091,10753,15361,16001,19841,21001,21401,24001,
%U A178070 25601,27961,37501,40961,43201,60101,62501,65537,69857,76001,76801,160001,162251,163841,307201,453377,524801,544001,670001,952001,976193,980801
%N A178070 Primes dividing repunits R(10^n) for some n.
%C A178070 Repunits are the numbers consisting entirely of 1's. The number represented by R(10^n) contains 10^n digits with all 1's. E.g., R(10^1) = 1111111111.
%C A178070 A prime p > 5 is here if the multiplicative order of 10 (mod p) is of the form 2^i*5^j, with i and j nonnegative.
%C A178070 Includes all terms > 5 of A077497. - _Robert Israel_, Nov 05 2024
%H A178070 Robert Israel, <a href="/A178070/b178070.txt">Table of n, a(n) for n = 1..110</a>
%H A178070 Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/glpxm1.pl">Known prime factors of Googolplexplex - 1</a>
%H A178070 Project Euler, <a href="https://projecteuler.net/problem=133">Problem 133: Repunit nonfactors</a>
%H A178070 Robert P. Munafo, <a href="https://mrob.com/pub/math/numbers-22.html#lp2_e008_37">Notable Properties of Specific Numbers</a>
%e A178070 17 divides R(10^4), so is in the sequence. - _Phil Carmody_, May 26 2011
%e A178070 Note that R(10^n) == 1 mod 3 for all n, so 3 is not a member. - _N. J. A. Sloane_, Jun 18 2014
%p A178070 filter:= proc(p) local v;
%p A178070   if not isprime(p) then return false fi;
%p A178070   v:= numtheory:-order(10,p);
%p A178070   v = 2^padic:-ordp(v,2) * 5^padic:-ordp(v,5)
%p A178070 end proc:
%p A178070 select(filter, [seq(i, i=7 .. 10^6, 2)]); # _Robert Israel_, Nov 05 2024
%t A178070 Select[Prime[Range[4, 100000]], Complement[First /@ FactorInteger[MultiplicativeOrder[10, #]], {2, 5}] == {} &] (* _T. D. Noe_, May 26 2011 *)
%o A178070 (PARI) g=10^30;forprime(p=7,1000000,z=znorder(Mod(10,p));if(gcd(z,g)==z,print1(p", "))) \\ _Phil Carmody_, May 26 2011
%o A178070 (PARI) upTo(lim)=my(v=List(),g=10^(log(lim)\log(2))); forprime(p=7,lim,if(g%znorder(Mod(10,p))==0, listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, May 26 2011
%Y A178070 Cf. A077497, A227246.
%K A178070 nonn
%O A178070 1,1
%A A178070 _Shashank Sharma_, May 19 2010, Aug 04 2010
%E A178070 Arbitrary limit removed and sequence extended by _Phil Carmody_, May 26 2011