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A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.

%I A238974 #16 Mar 26 2020 18:53:56
%S A238974 0,1,3,5,6,12,19,10,22,27,42,65,15,35,48,74,90,138,211,21,51,75,115,
%T A238974 84,156,238,189,288,438,665,28,70,108,165,130,240,365,268,324,492,746,
%U A238974 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
%N A238974 The size (the number of arcs) in the transitive closure of divisor lattice in canonical order.
%H A238974 Andrew Howroyd, <a href="/A238974/b238974.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238974 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.
%F A238974 T(n,k) = A238952(A063008(n,k)). - _Andrew Howroyd_, Mar 26 2020
%e A238974 Triangle T(n,k) begins:
%e A238974    0;
%e A238974    1;
%e A238974    3,  5;
%e A238974    6, 12, 19;
%e A238974   10, 22, 27,  42, 65;
%e A238974   15, 35, 48,  74, 90, 138, 211;
%e A238974   21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665;
%e A238974   ...
%p A238974 with(numtheory):
%p A238974 f:= n-> add(tau(d), d=divisors(n) minus {n}):
%p A238974 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238974     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238974 T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
%p A238974 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 26 2020
%o A238974 (PARI) \\ here b(n) is A238952.
%o A238974 b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)}
%o A238974 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238974 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238974 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 26 2020
%Y A238974 Cf. A238961 in canonical order.
%Y A238974 Cf. A063008, A238952.
%K A238974 nonn,tabf
%O A238974 0,3
%A A238974 _Sung-Hyuk Cha_, Mar 07 2014
%E A238974 Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 26 2020