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

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

Author

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

Keywords

• nonn
• fini

Comments

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)

Examples

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

Links

• 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

Crossrefs

Cf. A227246.