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

1,4

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).

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, d<n} A000005(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

Mathematica

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

Prog

(PARI) A238952(n) = sumdiv(n, d, (d<n)*numdiv(d)); \\ Antti Karttunen, Mar 07 2018, after Geoffrey Critzer's Mathematica-code.

Crossrefs

Cf. A000005, A007425, A062799.

Sequence in context: A002972 A324896 A029652 * A333976 A129510 A225656

Adjacent sequences: A238949 A238950 A238951 * A238953 A238954 A238955

Keyword

nonn

Author

Sung-Hyuk Cha, Mar 07 2014

Status

approved