A262535 - OEIS

%I #16 Oct 02 2015 16:55:41

%S 1,2,3,3,4,3,5,8,6,8,3,7,15,3,8,15,3,9,24,6,10,24,6,5,11,35,6,5,12,35,

%T 9,5,13,48,9,5,14,48,9,12,15,63,12,12,5,16,63,12,12,5,17,80,12,12,5,

%U 18,80,15,21,5,19,99,15,21,5,20,99,15,21,10,21,120,18,21,10,7,22,120,18,32,10,7,23,143,18,32,10,7,24,143,21,32,10,7,25,168,21,32,15,7,26,168,21,45,15,7,27,195,24,45,15,16

%N Triangle read by rows T(n,k) in which column k lists the partial sums of the k-th column of triangle A261699.

%C Conjecture: the sum of row n gives A078471(n), the sum of all odd divisors of all positive integers <= n.

%C Row n has length A003056(n) hence column k starts in row A000217(k).

%C Column 1 gives A000027.

%e Triangle begins:

%e 1;

%e 2;

%e 3,    3;

%e 4,    3;

%e 5,    8;

%e 6,    8,  3;

%e 7,   15,  3;

%e 8,   15,  3;

%e 9,   24,  6;

%e 10,  24,  6,  5;

%e 11,  35,  6,  5;

%e 12,  35,  9,  5;

%e 13,  48,  9,  5;

%e 14,  48,  9, 12;

%e 15,  63, 12, 12,  5;

%e 16,  63, 12, 12,  5;

%e 17,  80, 12, 12,  5;

%e 18,  80, 15, 21,  5;

%e 19,  99, 15, 21,  5;

%e 20,  99, 15, 21, 10;

%e 21, 120, 18, 21, 10,  7;

%e 22, 120, 18, 32, 10,  7;

%e 23, 143, 18, 32, 10,  7;

%e 24, 143, 21, 32, 10,  7;

%e 25, 168, 21, 32, 15,  7;

%e 26, 168, 21, 45, 15,  7;

%e 27, 195, 24, 45, 15, 16;

%e ...

%e For n = 6 the sum of all odd divisors of all positive integers <= 6 is (1) + (1) + (1 + 3) + (1) + (1 + 5) + (1 + 3) = 17. On the other hand the sum of the 6th row of triangle is 6 + 8 + 3 = 17 equaling the sum of all odd divisors of all positive integers <= 6.

%Y Cf. A000027, A000217, A001227, A003056, A060831, A078471, A236104, A237593, A261699.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Sep 24 2015