A253249 Number of nonempty chains in the divides relation on the divisors of n.

1, 3, 3, 7, 3, 11, 3, 15, 7, 11, 3, 31, 3, 11, 11, 31, 3, 31, 3, 31, 11, 11, 3, 79, 7, 11, 15, 31, 3, 51, 3, 63, 11, 11, 11, 103, 3, 11, 11, 79, 3, 51, 3, 31, 31, 11, 3, 191, 7, 31, 11, 31, 3, 79, 11, 79, 11, 11, 3, 175, 3, 11, 31, 127, 11, 51, 3, 31, 11, 51

Offset

1,2

Comments

For prime p, a(p)=3.

a(2^k) = 2^(k+1)-1.

For integers of the form n = p_1*p_2*...*p_k we have a(n) = A007047(k).

The value of a(n) depends only on the exponents in the prime factorization of n.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Formula

Dirichlet g.f.: zeta(s)^2*A(s) where A(s) is the Dirichlet g.f. for A074206. - Geoffrey Critzer, May 23 2018

Sum_{k=1..n} a(k) ~ -4*n^r / (r*Zeta'(r)), where r = A107311 = 1.728647238998183618135103... is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Jan 31 2019

a(n) = 4*A002033(n-1) - 1 for n > 1. - Geoffrey Critzer, Aug 19 2020

Example

a(10) = 11 because we have: {1}, {2}, {5}, {10}, {1|2}, {1|5}, {1|10}, {2|10}, {5|10}, {1|2|10}, {1|5|10}.

Maple

with(numtheory):
b:= proc(n) option remember: 1+ `if`(n=1, 0,
       add(b(d), d=divisors(n) minus {n}))
    end:
a:= n-> add(b(d), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 04 2015

Mathematica

Table[Total[Table[Length[Select[Subsets[Divisors[n], {k}], Apply[And, Map[Apply[Divisible, #] &, Partition[Reverse[#], 2, 1]]] &]], {k, 1, PrimeOmega[n] + 1}]], {n, 1, 100}]

Crossrefs

Cf. A002033, A007047, A074206, A107311.

Sequence in context: A135434 A204204 A164928 * A069949 A143275 A083262

Adjacent sequences: A253246 A253247 A253248 * A253250 A253251 A253252

Keyword

nonn

Author

Geoffrey Critzer, Jun 04 2015

Status

approved