

A019569


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


1



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
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OFFSET

1,3


COMMENTS

Let p(i) = ith 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 s1 scores into s segments. The scores and the bar divide the space above and below the bar into 2's compartments. In the ith compartment above the bar place the drawing of e_i and in the ith compartment below the bar place the drawing of k_i  k_{i1}.


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_{i1}) ) with k_0 = 0.


MAPLE

with(numtheory):
a:= proc(n) option remember; `if`(n<3, n1, (l> nops(l)+
add(a(l[i, 2])+a(pi(l[i, 1])`if`(i=1, 0, pi(l[i1, 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, n1, Function[l, Length[l] + Sum[a[l[[i, 2]]] + a[PrimePi[l[[i, 1]]]  If[i == 1, 0, PrimePi[l[[i1, 1]]]]], {i, 1, Length[l]}]][SortBy[FactorInteger[n], First]]];
Array[a, 100] (* JeanFrançois 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



