A087094 a(n) = smallest k such that (10^k-1)/9 == 0 mod prime(n)^2, or 0 if no such k exists.

0, 9, 0, 42, 22, 78, 272, 342, 506, 812, 465, 111, 205, 903, 2162, 689, 3422, 3660, 2211, 2485, 584, 1027, 3403, 3916, 9312, 404, 3502, 5671, 11772, 12656, 5334, 17030, 1096, 6394, 22052, 11325, 12246, 13203, 27722, 7439, 31862, 32580, 18145, 37056, 19306

Offset: 1

Author

Ray Chandler, Aug 10 2003

Keywords

• nonn

Comments

For a given a(n)>0, all of the values of k such that (10^k-1)/9=0 mod prime(n)^2 is given by the sequence a(n)*A000027, i.e. integral multiples of a(n). For example, for n=2, prime(2)=3, a(n)=9, the set of values of k for which (10^k-1)/9=0 mod 3^2 is 9*A000027=9,18,27,36,45,...

The union of the collection of sequences formed from the nonzero terms of a(n)*A000027, gives the values of k for which (10^k-1)/9 is not squarefree, see A046412. All of terms of the sequence a(n) are integer multiples of prime(n) for primes <1000 except for a(93)=486 where prime(93)=487. Conjecture: there are no 0 terms after a(3).

That conjecture is easily proved, for a(n) is just the multiplicative order of 10 modulo (prime(n))^2 for n>3. - Jeppe Stig Nielsen, Dec 28 2015

Examples

			a(2)=9 since 9 is least value of k for which (10^k-1)/9=0 mod 3^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000042, A046412, A084006, A084007.

Programs

Maple

0,9,0,seq(numtheory:-order(10,ithprime(i)^2), i=4..100); # Robert Israel, Dec 30 2015

PARI

a(n)=p=prime(n);10%p==0 && return(0);for(k=1,p^2,((10^k-1)/9) % p^2 == 0 && return(k));error() \\ Jeppe Stig Nielsen, Dec 28 2015

PARI

a(n)=p=prime(n);if(10%p==0, 0, 10%p==1, 9, znorder(Mod(10,p^2))) \\ Jeppe Stig Nielsen, Dec 28 2015

Formula

For n>3, a(n) = A084680(prime(n)^2) = A084680(A001248(n)), Jeppe Stig Nielsen, Dec 28 2015