A096085 a(n) is the least positive integer such that for 1 <= k <= n, the concatenation of the k terms a(n-k+1) through a(n) is a multiple of k.

1, 2, 6, 4, 20, 60, 340, 920, 600, 1000, 14240, 31560, 100120, 140960, 315960, 314800, 7523840, 1020480, 109764640, 66757520, 23256960, 2200720, 1260893360, 1059221040, 16184204800, 34159566800, 18162880800, 68345405200

Offset

1,2

Comments

It suffices to check this for prime-power values of k. To see this, let k = a*b, where a and b are relatively prime. The concatenation of k terms can be viewed as a concatenation of a numbers, each of which is the concatenation of b terms. The first a-1 of these were previously chosen to be multiples of b, so if the last is also a multiple of b, then the concatenation of all k terms is a multiple of b. By the same argument, the concatenation of all k terms is a multiple of a and since a and b are relatively prime, it is a multiple of k. - David Wasserman, May 21 2007

a(n) exists for all n, because the Chinese Remainder Theorem shows that a d-digit solution must exist if 9*10^(d-1) >= A003418(n). - David Wasserman, May 21 2007

Links

Table of n, a(n) for n=1..28.

Example

12, 26, 64 and 420 are multiples of 2.
126, 264, 6420 and 42060 are multiples of 3.
1264, 26420, 642060 and 42060340 are multiples of 4.
126420, 2642060, 642060340 and 42060340920 are multiples of 5.

Crossrefs

Sequence in context: A069875 A202962 A019088 * A305745 A285988 A218973

Adjacent sequences: A096082 A096083 A096084 * A096086 A096087 A096088

Keyword

base,nonn,less

Author

Amarnath Murthy, Jun 22 2004

Extensions

Edited and extended by David Wasserman, May 21 2007

Status

approved