A229964 Number of pairs of integers q1, q2 with 1 < q1 < q2 < n such that if we randomly pick an integer in {1, ..., n}, the event of being divisible by q1 is independent of being divisible by q2.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 3, 0, 3, 2, 1, 0, 4, 0, 5, 1, 3, 0, 8, 0, 4, 3, 4, 0, 10, 0, 7, 3, 5, 2, 9, 0, 6, 4, 9, 0, 13, 0, 12, 6, 6, 0, 16, 0, 9, 6, 9, 0, 14, 1, 12, 3, 8, 0, 25, 0, 12, 10, 11, 4, 17, 0, 12, 7, 17, 0, 25, 0, 14, 12, 14, 2, 21, 0, 21, 5

Offset

1,12

Links

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Rosemary Sullivan and Neil Watling, Independent Divisibility Pairs on the Set of Integers from 1 to N, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A65, 2013.

Example

If n = 12, then q1 = 2 and q2 = 5 satisfy the condition as the probability of an integer in {1, ..., 12} being divisible by 2 is 1/2, by 5 is 1/6, and by both 2 and 5 is 1/12.

Prog

(Sage)
def A229964(n) : return sum(sum(dprob(q1, n) * dprob(q2, n) == dprob(lcm(q1, q2), n) for q2 in range(q1+1, n)) for q1 in n.divisors() if q1 not in [1, n])
def dprob(q, n) : return (n // q)/n

Crossrefs

The n such that a(n) = m for various m are given by: m=0, A166684; m=1, A229965; m=2, A082663; m=3, A229966; m=4, A229967.

Sequence in context: A105824 A171911 A180193 * A309722 A070298 A024938

Adjacent sequences: A229961 A229962 A229963 * A229965 A229966 A229967

Keyword

nonn

Author

Eric M. Schmidt, Oct 04 2013

Status

approved