A238974 - OEIS

%I #16 Mar 26 2020 18:53:56

%S 0,1,3,5,6,12,19,10,22,27,42,65,15,35,48,74,90,138,211,21,51,75,115,

%T 84,156,238,189,288,438,665,28,70,108,165,130,240,365,268,324,492,746,

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

%H Andrew Howroyd, <a href="/A238974/b238974.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)

%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.

%F T(n,k) = A238952(A063008(n,k)). - _Andrew Howroyd_, Mar 26 2020

%e Triangle T(n,k) begins:

%e    0;

%e    1;

%e    3,  5;

%e    6, 12, 19;

%e   10, 22, 27,  42, 65;

%e   15, 35, 48,  74, 90, 138, 211;

%e   21, 51, 75, 115, 84, 156, 238, 189, 288, 438, 665;

%e   ...

%p with(numtheory):

%p f:= n-> add(tau(d), d=divisors(n) minus {n}):

%p b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->

%p     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):

%p T:= n-> map(x-> f(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:

%p seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 26 2020

%o (PARI) \\ here b(n) is A238952.

%o b(n) = {sumdivmult(n, d, numdiv(d)) - numdiv(n)}

%o N(sig)={prod(k=1, #sig, prime(k)^sig[k])}

%o Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}

%o { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 26 2020

%Y Cf. A238961 in canonical order.

%Y Cf. A063008, A238952.

%K nonn,tabf

%O 0,3

%A _Sung-Hyuk Cha_, Mar 07 2014

%E Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 26 2020