A238971 - OEIS

%I #19 Apr 24 2020 11:44:09

%S 0,1,1,2,2,3,4,2,4,4,6,8,3,5,6,8,9,12,16,3,6,7,10,8,12,16,13,18,24,32,

%T 4,7,9,12,10,15,20,16,18,24,32,27,36,48,64,4,8,10,14,12,18,24,12,20,

%U 22,30,40,24,32,36,48,64,40,54,72,96,128

%N The number of nodes at odd level in divisor lattice in canonical order.

%H Andrew Howroyd, <a href="/A238971/b238971.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 From _Andrew Howroyd_, Mar 25 2020: (Start)

%F T(n,k) = A056924(A063008(n,k)).

%F T(n,k) = A238963(n,k) - A238970(n,k).

%F T(n,k) = floor(A238963(n,k)/2). (End)

%e Triangle T(n,k) begins:

%e   0;

%e   1;

%e   1, 2;

%e   2, 3, 4;

%e   2, 4, 4,  6, 8;

%e   3, 5, 6,  8, 9, 12, 16;

%e   3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32;

%e   ...

%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-> floor(numtheory[tau](mul(ithprime(i)

%p         ^x[i], i=1..nops(x)))/2), b(n$2))[]:

%p seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 25 2020

%o (PARI)

%o b(n)={numdiv(n)\2}

%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 25 2020

%Y Cf. A238958 in canonical order.

%Y Cf. A056924, A063008, A238963, A238970.

%K nonn,tabf

%O 0,4

%A _Sung-Hyuk Cha_, Mar 07 2014

%E Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020