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A238952 The size (the number of arcs) in the transitive closure of divisor lattice D(n). 2
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 (list; graph; refs; listen; history; text; internal format)
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

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 * A129510 A225656 A087913

Adjacent sequences:  A238949 A238950 A238951 * A238953 A238954 A238955

KEYWORD

nonn

AUTHOR

Sung-Hyuk Cha, Mar 07 2014

STATUS

approved

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Last modified October 23 09:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)