cp's OEIS Frontend

A238970 The number of nodes at even level in divisor lattice in canonical order.

%I A238970 #25 Jan 30 2025 15:38:55
%S A238970 1,1,2,2,2,3,4,3,4,5,6,8,3,5,6,8,9,12,16,4,6,8,10,8,12,16,14,18,24,32,
%T A238970 4,7,9,12,10,15,20,16,18,24,32,27,36,48,64,5,8,11,14,12,18,24,13,20,
%U A238970 23,30,40,24,32,36,48,64,41,54,72,96,128
%N A238970 The number of nodes at even level in divisor lattice in canonical order.
%H A238970 Andrew Howroyd, <a href="/A238970/b238970.txt">Table of n, a(n) for n = 0..2713</a> (rows 0..20)
%H A238970 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 A238970 From _Andrew Howroyd_, Mar 25 2020: (Start)
%F A238970 T(n,k) = A038548(A063008(n,k)).
%F A238970 T(n,k) = A238963(n,k) - A238971(n,k).
%F A238970 T(n,k) = ceiling(A238963(n,k)/2). (End)
%e A238970 Triangle T(n,k) begins:
%e A238970   1;
%e A238970   1;
%e A238970   2, 2;
%e A238970   2, 3, 4;
%e A238970   3, 4, 5,  6, 8;
%e A238970   3, 5, 6,  8, 9, 12, 16;
%e A238970   4, 6, 8, 10, 8, 12, 16, 14, 18, 24, 32;
%e A238970   ...
%p A238970 b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
%p A238970     [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
%p A238970 T:= n-> map(x-> ceil(numtheory[tau](mul(ithprime(i)
%p A238970         ^x[i], i=1..nops(x)))/2), b(n$2))[]:
%p A238970 seq(T(n), n=0..9);  # _Alois P. Heinz_, Mar 25 2020
%t A238970 A063008row[n_] := Product[Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n];
%t A238970 A038548[n_] := Ceiling[DivisorSigma[0, n]/2];
%t A238970 T[n_] := A038548 /@ A063008row[n];
%t A238970 Table[T[n], {n, 0, 9}] // Flatten (* _Jean-François Alcover_, Jan 30 2025 *)
%o A238970 (PARI) \\ here b(n) is A038548.
%o A238970 b(n)={ceil(numdiv(n)/2)}
%o A238970 N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
%o A238970 Row(n)={apply(s->b(N(s)), vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
%o A238970 { for(n=0, 8, print(Row(n))) } \\ _Andrew Howroyd_, Mar 25 2020
%Y A238970 Cf. A238957 in canonical order.
%Y A238970 Leftmost column gives A008619.
%Y A238970 Last terms of rows give A011782.
%Y A238970 Cf. A038548, A063008, A238963, A238971.
%K A238970 nonn,tabf
%O A238970 0,3
%A A238970 _Sung-Hyuk Cha_, Mar 07 2014
%E A238970 Offset changed and terms a(50) and beyond from _Andrew Howroyd_, Mar 25 2020