id	title	challengeType	forumTopicId	dashedName
5900f4b21000cf542c50ffc5	Problem 326: Modulo Summations	5	301983	problem-326-modulo-summations

--description--

Let a_n be a sequence recursively defined by: a_1 = 1, \displaystyle a_n = \left(\sum_{k = 1}^{n - 1} k \times a_k\right)\bmod n.

So the first 10 elements of a_n are: 1, 1, 0, 3, 0, 3, 5, 4, 1, 9.

Let f(N, M) represent the number of pairs (p, q) such that:

 1 \le p \le q \le N \\; \text{and} \\; \left(\sum_{i = p}^q a_i\right)\bmod M = 0

It can be seen that f(10, 10) = 4 with the pairs (3,3), (5,5), (7,9) and (9,10).

You are also given that f({10}^4, {10}^3) = 97\\,158.

Find f({10}^{12}, {10}^6).

--hints--

moduloSummations() should return 1966666166408794400.

assert.strictEqual(moduloSummations(), 1966666166408794400);

function moduloSummations() {

  return true;
}

moduloSummations();

// solution required