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A238971 The number of nodes at odd level in divisor lattice in canonical order.

%I A238971 #19 Apr 24 2020 11:44:09
%S A238971 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 A238971 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 A238971 22,30,40,24,32,36,48,64,40,54,72,96,128
%N A238971 The number of nodes at odd level in divisor lattice in canonical order.
%H A238971 Andrew Howroyd, <a href="/A238971/b238971.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238971 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 A238971 From _Andrew Howroyd_, Mar 25 2020: (Start)
%F A238971 T(n,k) = A056924(A063008(n,k)).
%F A238971 T(n,k) = A238963(n,k) - A238970(n,k).
%F A238971 T(n,k) = floor(A238963(n,k)/2). (End)
%e A238971 Triangle T(n,k) begins:
%e A238971   0;
%e A238971   1;
%e A238971   1, 2;
%e A238971   2, 3, 4;
%e A238971   2, 4, 4,  6, 8;
%e A238971   3, 5, 6,  8, 9, 12, 16;
%e A238971   3, 6, 7, 10, 8, 12, 16, 13, 18, 24, 32;
%e A238971   ...
%p A238971 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238971     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238971 T:= n-> map(x-> floor(numtheory[tau](mul(ithprime(i)
%p A238971         ^x[i], i=1..nops(x)))/2), b(n$2))[]:
%p A238971 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 25 2020
%o A238971 (PARI)
%o A238971 b(n)={numdiv(n)\2}
%o A238971 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238971 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238971 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020
%Y A238971 Cf. A238958 in canonical order.
%Y A238971 Cf. A056924, A063008, A238963, A238970.
%K A238971 nonn,tabf
%O A238971 0,4
%A A238971 _Sung-Hyuk Cha_, Mar 07 2014
%E A238971 Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020