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%I #39 Feb 21 2017 08:16:26
%S 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 15,14,12,8,16,17,16,15,12,9,18,19,18,16,15,10,20,18,14,21,20,11,22,
%U 23,22,21,20,18,16,12,24,20,25,24,13,26,24,18,27,26,24
%N Triangle read by rows: T(n,k) = n minus the k-th proper divisor of n.
%C 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 Alois P. Heinz, <a href="/A208460/b208460.txt">Rows n = 2..1540, flattened</a>
%H 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 T(n,k) = n - A027751(n,k).
%e Written as a triangle starting from n = 2:
%e 1;
%e 2;
%e 3, 2;
%e 4;
%e 5, 4, 3;
%e 6;
%e 7, 6, 4;
%e 8, 6;
%e 9, 8, 5;
%e 10;
%e 11, 10, 9, 8, 6;
%e 12;
%p with (numtheory):
%p T:= n-> map(x-> n-x, sort([(divisors(n) minus {n})[]]))[]:
%p seq (T(n), n=2..50); # _Alois P. Heinz_, Apr 11 2012
%t T[n_] := Most[n-Divisors[n]]; Table[T[n], {n, 2, 50}] // Flatten (* _Jean-François Alcover_, Feb 21 2017 *)
%Y Column 1 is A000027. Row n has length A032741(n). Row sums give the positives A094471. Right border is A060681.
%Y Cf. A000005, A027750, A027751.
%K nonn,tabf
%O 2,2
%A _Omar E. Pol_, Feb 28 2012
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