A227246 -id:A227246 - cp's OEIS frontend

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

11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, 1201, 1409, 1601, 3541, 4001, 4801, 5051, 9091, 10753, 15361, 16001, 19841, 21001, 21401, 24001, 25601, 27961, 37501, 40961, 43201, 60101, 62501, 65537, 69857, 76001, 76801, 160001, 162251, 163841, 307201, 453377, 524801, 544001, 670001, 952001, 976193, 980801

Offset: 1

Shashank Sharma, May 19 2010, Aug 04 2010

nonn

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.
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.
Includes all terms > 5 of A077497. - Robert Israel, Nov 05 2024

			17 divides R(10^4), so is in the sequence. - _Phil Carmody_, May 26 2011
Note that R(10^n) == 1 mod 3 for all n, so 3 is not a member. - _N. J. A. Sloane_, Jun 18 2014

Robert Israel, Table of n, a(n) for n = 1..110
Dario Alejandro Alpern, Known prime factors of Googolplexplex - 1
Project Euler, Problem 133: Repunit nonfactors
Robert P. Munafo, Notable Properties of Specific Numbers

Cf. A077497, A227246.

filter:= proc(p) local v;
  if not isprime(p) then return false fi;
  v:= numtheory:-order(10,p);
  v = 2^padic:-ordp(v,2) * 5^padic:-ordp(v,5)
end proc:
select(filter, [seq(i, i=7 .. 10^6, 2)]); # Robert Israel, Nov 05 2024

Select[Prime[Range[4, 100000]], Complement[First /@ FactorInteger[MultiplicativeOrder[10, #]], {2, 5}] == {} &] (* T. D. Noe, May 26 2011 *)

g=10^30;forprime(p=7,1000000,z=znorder(Mod(10,p));if(gcd(z,g)==z,print1(p", "))) \\ Phil Carmody, May 26 2011

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

Arbitrary limit removed and sequence extended by Phil Carmody, May 26 2011

A278835 Prime factors (counting multiplicity) of 10^10^10^10^2 - 1.

Original entry on oeis.org

3, 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, 1201, 1409, 1601, 3541, 4001, 4801, 5051, 9091, 10753, 15361, 16001, 19841, 21001, 21401, 24001, 25601, 27961, 37501, 40961, 43201, 60101, 62501, 65537, 69857, 76001, 76801, 160001, 162251, 163841, 307201, 453377, 524801, 544001, 670001, 952001, 976193, 980801

Offset: 1

Robert Munafo and Robert G. Wilson v, Nov 28 2016

From Jon E. Schoenfield, Dec 02 2016, paraphrasing information from the Munafo link: (Start)
The decimal expansion of 10^10^10^10^2 - 1 would be 1 googolplex digits long, with each digit a 9. Many factors of this number can be identified using simple facts of modular arithmetic.
Since its digits are all 9's, it is divisible by 9=3*3. Since its digits are all 9's and the number of digits is even, it is divisible by 99 (as are 9999=99*101, 999999=99*10101, 99999999=99*1010101, etc.), and thus divisible by 11.
By the same principle, it is divisible by 9999, 99999, 99999999, and by any other number whose decimal expansion consists of k 9's where k is of the form 2^a * 5^b, where a and b are nonnegative integers up to 10^100 (see A003592) and all their divisors. Additional factors can be found using Fermat's Little Theorem.
Consequently, a large number of factors of 10^10^10^10^2 - 1 are known. (End)

			10^10^10^10^2 - 1 = 10^10^10^100 - 1 = 999...999 (a total of a googolplex of nines).

Robert G. Wilson v, Table of n, a(n) for n = 1..587
Dario Alejandro Alpern, Known prime factors of Googolduplex - 1
Dario Alejandro Alpern, Known 1-digit prime factors of Googolduplex - 1
Robert P. Munafo, Notable Properties of Specific Numbers

Cf. A227246.

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

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