A172372 Least number k such that the n-th prime not dividing 10 (A004139(n)) divides the repunit (10^k-1)/9.

3, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79, 110

Offset: 1

Author

Michel Lagneau, Feb 01 2010

Keywords

• nonn

Comments

If p is an odd prime different from 5, then p divides an infinite number of terms of the sequence of repunits {1, 11, 111, 1111, ... }. The proof is elementary: let p be such a prime. If p = 3, then 3 divides (10^3-1)/9 = 111. Otherwise, take k = (10^p - 1)/9; by the Fermat theorem, 10^(p-1) == 1 (mod p), so p divides (10^(p-1)-1); since p is relatively prime to 9, it divides k. Trivially, if p divides any k digit repunit, it divides the k*m digit repunit as well.

Essentially the same as A002371. - T. D. Noe, Apr 11 2012

Examples

			3 divides 111, but not 1 or 11, so a(1) = 3.
7 divides 111111 but not 1, 11, 111, 1111, or 11111, so a(2) = 6.
		

References

• David Wells, The Factors of the Repunits 11 through R_40, The Penguin Dictionary of Curious and Interesting Numbers, p. 219 Penguin 1986.
• David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1997.
• David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.

Links

• Project Euler, Problem 129: Repunit divisibility
• S. S. Wagstaff, Jr., The Cunningham Project
• Eric Weisstein's World of Mathematics, Repunit.

Crossrefs

Cf. A002275 (repunits), A002371 (period of decimal expansion of 1/prime(n)), A004139 (odd primes excluding 5), A095250 (11111111... (n times) mod n).

Programs

PARI

a(n) = {k=1; p = if(n>1, prime(n+2), 3); while((10^k-1)/9 % p, k++); k;} \\ Michel Marcus, May 25 2014

Extensions

Corrected and edited by Michel Lagneau, Apr 25 2010

Term 6 between terms 44 and 96 doesn't belong to the sequence. The same for term 43 between terms 43 and 178. Corrected and edited by Krzysztof Wojtas, May 25 2014