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A208460 Triangle read by rows: T(n,k) = n minus the k-th proper divisor of n.

%I A208460 #39 Feb 21 2017 08:16:26
%S A208460 1,2,3,2,4,5,4,3,6,7,6,4,8,6,9,8,5,10,11,10,9,8,6,12,13,12,7,14,12,10,
%T A208460 15,14,12,8,16,17,16,15,12,9,18,19,18,16,15,10,20,18,14,21,20,11,22,
%U A208460 23,22,21,20,18,16,12,24,20,25,24,13,26,24,18,27,26,24
%N A208460 Triangle read by rows: T(n,k) = n minus the k-th proper divisor of n.
%C A208460 Conjecture: one of the divisors of T(n,k) is also the k-th divisor of n. In a diagram of the structure of divisors of the natural numbers (see link) the mentioned divisors of the elements of row n are located on a straight line to 45 degrees from the vertical straight line that contains the divisors of n, therefore the divisors of n are predictable.
%H A208460 Alois P. Heinz, <a href="/A208460/b208460.txt">Rows n = 2..1540, flattened</a>
%H A208460 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv04.jpg">Illustration of the structure of divisors of the natural numbers, for n = 1..16</a>
%F A208460 T(n,k) = n - A027751(n,k).
%e A208460 Written as a triangle starting from n = 2:
%e A208460 1;
%e A208460 2;
%e A208460 3, 2;
%e A208460 4;
%e A208460 5, 4, 3;
%e A208460 6;
%e A208460 7, 6, 4;
%e A208460 8, 6;
%e A208460 9, 8, 5;
%e A208460 10;
%e A208460 11, 10, 9, 8, 6;
%e A208460 12;
%p A208460 with (numtheory):
%p A208460 T:= n-> map(x-> n-x, sort([(divisors(n) minus {n})[]]))[]:
%p A208460 seq (T(n), n=2..50); # _Alois P. Heinz_, Apr 11 2012
%t A208460 T[n_] := Most[n-Divisors[n]]; Table[T[n], {n, 2, 50}] // Flatten (* _Jean-François Alcover_, Feb 21 2017 *)
%Y A208460 Column 1 is A000027. Row n has length A032741(n). Row sums give the positives A094471. Right border is A060681.
%Y A208460 Cf. A000005, A027750, A027751.
%K A208460 nonn,tabf
%O A208460 2,2
%A A208460 _Omar E. Pol_, Feb 28 2012