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A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

%I A207375 #49 Apr 07 2025 09:26:50
%S A207375 1,1,2,1,3,2,1,5,2,3,1,7,2,4,3,2,5,1,11,3,4,1,13,2,7,3,5,4,1,17,3,6,1,
%T A207375 19,4,5,3,7,2,11,1,23,4,6,5,2,13,3,9,4,7,1,29,5,6,1,31,4,8,3,11,2,17,
%U A207375 5,7,6,1,37,2,19,3,13,5,8,1,41,6,7,1,43
%N A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.
%C A207375 If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.
%C A207375 If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.
%C A207375 Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).
%C A207375 Column 1 gives A033676. Right border gives A033677. - _Omar E. Pol_, Feb 26 2019
%C A207375 The conjecture 1 follows from Bertrand's Postulate. - _Charles R Greathouse IV_, Feb 11 2022
%C A207375 Row products give A097448. - _Omar E. Pol_, Feb 17 2022
%H A207375 Alois P. Heinz, <a href="/A207375/b207375.txt">Rows n = 1..5000</a>
%H A207375 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of the divisors of the first 12 positive integers</a>
%e A207375 Array begins:
%e A207375   1;
%e A207375   1,  2;
%e A207375   1,  3;
%e A207375   2;
%e A207375   1,  5;
%e A207375   2,  3;
%e A207375   1,  7;
%e A207375   2,  4;
%e A207375   3;
%e A207375   2,  5;
%e A207375   1, 11;
%e A207375   3,  4;
%e A207375   1, 13;
%e A207375 ...
%t A207375 A207375row[n_] := ArrayPad[#, -Floor[(Length[#] - 1)/2]] & [Divisors[n]];
%t A207375 Array[A207375row, 50] (* _Paolo Xausa_, Apr 07 2025 *)
%Y A207375 Row n has length A169695(n).
%Y A207375 Row sums give A207376.
%Y A207375 Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.
%K A207375 nonn,tabf
%O A207375 1,3
%A A207375 _Omar E. Pol_, Feb 23 2012