A330757 Let d(1) < d(2) < ... < d(q) denote the divisors of n; a(n) is the number of elements of the set { d(1)/d(2), d(2)/d(3), ..., d(q-1)/d(q) }.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 1, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 6, 1, 2, 3, 1, 2, 3, 1, 2, 2, 4, 1, 4, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 1, 5, 2, 2, 2

Offset

1,6

Comments

This sequence is a variant of A060682; here we consider the quotients, there the differences of consecutive divisors.

The sequence is unbounded since a(n!) >= n-1 for any n > 0.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = 1 iff n is a prime power (A246655).

Example

For n = 42:
- the divisors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42,
- the corresponding quotients are: 1/2, 2/3, 1/2, 6/7, 1/2, 2/3, 1/2,
- which corresponds to the set { 1/2, 2/3, 6/7 },
- hence a(42) = 3.

Prog

(PARI) a(n) = my (d=divisors(n)); #Set(vector(#d-1, k, d[k]/d[k+1]))

Crossrefs

Cf. A060682, A246655.

Sequence in context: A327402 A123529 A140747 * A322373 A332288 A335450

Adjacent sequences: A330754 A330755 A330756 * A330758 A330759 A330760

Keyword

nonn

Author

Rémy Sigrist, Dec 29 2019

Status

approved