A019569 Number of bar segments in a certain way of representing the integers graphically.

0, 1, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 3, 5, 4, 5, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 5, 4, 5, 4, 4, 4, 4, 6, 6, 4, 4, 5, 6, 5, 5, 5, 4, 4, 5, 5, 3, 6, 5, 5, 5, 6, 4, 5, 5, 5, 6, 4, 6, 4, 5, 5, 5, 4

Offset

1,3

Comments

Let p(i) = i-th prime. Let n = Product_{i=1..s} p(k_i)^e_i with k_1 < k_2 < ... < k_s. The drawing of n=1 is a blank space. The drawing of n > 1 is arranged around a horizontal bar divided by s-1 scores into s segments. The scores and the bar divide the space above and below the bar into 2's compartments. In the i-th compartment above the bar place the drawing of e_i and in the i-th compartment below the bar place the drawing of k_i - k_{i-1}.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Sean A. Irvine, Java program (github)

Tyler Pierce, A way of drawing natural numbers

Formula

a(1) = 0; a(2) = 1; a(n) = s + Sum_{i=1..s} ( a(e_i) + a(k_i - k_{i-1}) ) with k_0 = 0.

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n<3, n-1, (l-> nops(l)+
      add(a(l[i, 2])+a(pi(l[i, 1])-`if`(i=1, 0, pi(l[i-1, 1]))),
      i=1..nops(l)))(sort(ifactors(n)[2], (x, y)-> x[1]<y[1])))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 26 2019

Mathematica

a[n_] := a[n] = If[n<3, n-1, Function[l, Length[l] + Sum[a[l[[i, 2]]] + a[PrimePi[l[[i, 1]]] - If[i == 1, 0, PrimePi[l[[i-1, 1]]]]], {i, 1, Length[l]}]][SortBy[FactorInteger[n], First]]];
Array[a, 100] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Crossrefs

Sequence in context: A202472 A235613 A322418 * A003434 A330808 A097849

Adjacent sequences: A019566 A019567 A019568 * A019570 A019571 A019572

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

Formula corrected by Sean A. Irvine, Mar 26 2019

Status

approved