|
%I #17 Jul 08 2023 20:57:32
%S 3,3,11,17,41,73,101,137,251,257,271,353,401,449,641,751,1201,1409,
%T 1601,3541,4001,4801,5051,9091,10753,15361,16001,19841,21001,21401,
%U 24001,25601,27961,37501,40961,43201,60101,62501,65537,69857,76001,76801,160001,162251,163841,307201,453377,524801,544001,670001,952001,976193,980801
%N Prime factors (counting multiplicity) of 10^10^10^10^2 - 1.
%C From _Jon E. Schoenfield_, Dec 02 2016, paraphrasing information from the Munafo link: (Start)
%C 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.
%C 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.
%C 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.
%C Consequently, a large number of factors of 10^10^10^10^2 - 1 are known. (End)
%H Robert G. Wilson v, <a href="/A278835/b278835.txt">Table of n, a(n) for n = 1..587</a>
%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/glpxm1.pl">Known prime factors of Googolduplex - 1</a>
%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/glpxm1.pl?digits=1"> Known 1-digit prime factors of Googolduplex - 1</a>
%H Robert P. Munafo, <a href="http://mrob.com/pub/math/numbers-22.html#lp2_e008_37"> Notable Properties of Specific Numbers</a>
%e 10^10^10^10^2 - 1 = 10^10^10^100 - 1 = 999...999 (a total of a googolplex of nines).
%Y Cf. A227246.
%K nonn,fini
%O 1,1
%A _Robert Munafo_ and _Robert G. Wilson v_, Nov 28 2016
|
