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The size (the number of arcs) in the transitive closure of divisor lattice D(n).
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%I #41 Nov 07 2024 21:59:21

%S 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,

%T 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,

%U 42,1,5,12,21,5,19,1,12,5,19,1,48,1,5,12,12,5

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

%C 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

%C 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

%H Antti Karttunen, <a href="/A238952/b238952.txt">Table of n, a(n) for n = 1..65537</a>

%H S.-H. Cha, E. G. DuCasse, and L. V. Quintas, <a href="http://arxiv.org/abs/1405.5283">Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures</a>, arXiv:1405.5283 [math.NT], 2014 (see 13th line in Table 1).

%F 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

%F a(n) = Sum_{d|n, d<n} A000005(d). - _Antti Karttunen_, Mar 08 2018, after _Geoffrey Critzer_'s Mathematica-code.

%F 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

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

%o (PARI) A238952(n) = sumdiv(n, d, (d<n)*numdiv(d)); \\ _Antti Karttunen_, Mar 07 2018, after _Geoffrey Critzer_'s Mathematica-code.

%Y Cf. A000005, A007425, A062799.

%K nonn,changed

%O 1,4

%A _Sung-Hyuk Cha_, Mar 07 2014