A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.
0, 1, 3, 5, 6, 12, 19, 10, 22, 27, 42, 65, 15, 35, 48, 74, 90, 138, 211, 21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665, 28, 70, 108, 165, 130, 240, 365, 268, 324, 492, 746, 594, 900, 1362, 2059, 36, 92, 147, 224, 186, 342, 519, 200, 410, 495, 750, 1135, 552, 836, 1008, 1524, 2302, 1215, 1836, 2772, 4182, 6305
Offset: 0
Examples
Triangle T(n,k) begins: 0; 1; 3, 5; 6, 12, 19; 10, 22, 27, 42, 65; 15, 35, 48, 74, 90, 138, 211; 21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665; ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
- 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.
Programs
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Maple
with(numtheory): f:= n-> add(tau(d), d=divisors(n) minus {n}): b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x-> [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]): T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]: seq(T(n), n=0..9); # Alois P. Heinz, Mar 26 2020 -
PARI
\\ here b(n) is A238952. b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)} N(sig)={prod(k=1, #sig, prime(k)^sig[k])} Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))} { for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 26 2020
Formula
Extensions
Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 26 2020