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Sequences with nice divisibility properties

Published on Saturday, 11th April 2015, 07:00 pm; Solved by 334;
Difficulty rating: 55%

Problem 511

Let Seq(n,k) be the number of positive-integer sequences {ai}1≤i≤n of length n such that:

  • n is divisible by ai for 1 ≤ i ≤ n, and
  • n + a1 + a2 + ... + an is divisible by k.

Examples:

Seq(3,4) = 4, and the 4 sequences are:
{1, 1, 3}
{1, 3, 1}
{3, 1, 1}
{3, 3, 3}

Seq(4,11) = 8, and the 8 sequences are:
{1, 1, 1, 4}
{1, 1, 4, 1}
{1, 4, 1, 1}
{4, 1, 1, 1}
{2, 2, 2, 1}
{2, 2, 1, 2}
{2, 1, 2, 2}
{1, 2, 2, 2}

The last nine digits of Seq(1111,24) are 840643584.

Find the last nine digits of Seq(1234567898765,4321).