A238952 The size (the number of arcs) in the transitive closure of divisor lattice D(n).

0, 1, 1, 3, 1, 5, 1, 6, 3, 5, 1, 12, 1, 5, 5, 10, 1, 12, 1, 12, 5, 5, 1, 22, 3, 5, 6, 12, 1, 19, 1, 15, 5, 5, 5, 27, 1, 5, 5, 22, 1, 19, 1, 12, 12, 5, 1, 35, 3, 12, 5, 12, 1, 22, 5, 22, 5, 5, 1, 42, 1, 5, 12, 21, 5, 19, 1, 12, 5, 19, 1, 48, 1, 5, 12, 12, 5

Offset: 1

Author

Sung-Hyuk Cha, Mar 07 2014

Keywords

• nonn

Comments

a(n) is the number of ordered factorizations of n = r*s*t such that t is not equal to 1. For example: a(4)=3 because we have: 1*1*4, 1*2*2, and 2*1*2. Cf. A007425. - Geoffrey Critzer, Jan 01 2015

Number of pairs (d1, d2) of divisors of n such that d1<=d2, d1|n, d2|n, d1|d2 and d1 + d2 <= n. For example, a(8) has 6 divisor pairs (1,1), (1,2), (1,4), (2,2), (2,4) and (4,4). - Wesley Ivan Hurt, May 01 2021

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014 (see 13th line in Table 1).

Crossrefs

Cf. A000005, A007425, A062799.

Programs

Maple

with(numtheory): seq(add(tau(d), d in divisors(n) minus {n}), n=1..80); # Ridouane Oudra, Apr 26 2025

Mathematica

Table[Map[DivisorSigma[0, #] &, Drop[Divisors[n], -1]] // Total, {n, 1, 77}] (* Geoffrey Critzer, Jan 01 2015 *)

PARI

A238952(n) = sumdiv(n, d, (dAntti Karttunen, Mar 07 2018, after Geoffrey Critzer's Mathematica-code.

Formula

Conjecture: a(n) = Sum_{i=1..floor(n/2)} d(i) * (floor(n/i) - floor((n-1)/i)), where d(n) is the number of divisors of n. - Wesley Ivan Hurt, Dec 21 2017

a(n) = Sum_{d|n, dA000005(d). - Antti Karttunen, Mar 08 2018, after Geoffrey Critzer's Mathematica-code.

G.f.: Sum_{k>=1} (d(k) - 1)*x^k/(1 - x^k), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Sep 11 2018

a(n) = A007425(n) - A000005(n). - Ridouane Oudra, Apr 26 2025