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Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.
43

%I #41 Jul 25 2024 06:53:38

%S 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 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 5,7,6,1,37,2,19,3,13,5,8,1,41,6,7,1,43

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

%C 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 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 Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).

%C Column 1 gives A033676. Right border gives A033677. - _Omar E. Pol_, Feb 26 2019

%C The conjecture 1 follows from Bertrand's Postulate. - _Charles R Greathouse IV_, Feb 11 2022

%C Row products give A097448. - _Omar E. Pol_, Feb 17 2022

%H Alois P. Heinz, <a href="/A207375/b207375.txt">Rows n = 1..5000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">Illustration of the divisors of the first 12 positive integers</a>

%e Array begins:

%e 1;

%e 1, 2;

%e 1, 3;

%e 2;

%e 1, 5;

%e 2, 3;

%e 1, 7;

%e 2, 4;

%e 3;

%e 2, 5;

%e 1, 11;

%e 3, 4;

%e 1, 13;

%e ...

%Y Row n has length A169695(n).

%Y Row sums give A207376.

%Y Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Feb 23 2012