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A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.

%I A212120 #44 Oct 26 2024 22:58:18
%S A212120 1,3,5,7,1,9,1,11,3,13,3,15,5,17,5,1,19,7,1,21,7,1,23,9,3,25,9,3,27,
%T A212120 11,3,29,11,5,31,13,5,1,33,13,5,1,35,15,7,1,37,15,7,1,39,17,7,3,41,17,
%U A212120 9,3,43,19,9,3,45,19,9,3,47,21,11,5,49,21,11,5,1
%N A212120 Triangle read by rows T(n,k), n>=1, k>=1, where T(n,k) is the sum of the divisors d of n with min(d, n/d) = k.
%C A212120 Column k lists the odd numbers repeated k times starting in row k^2.
%C A212120 1 together with the first differences of the row sums give the divisor function A000005.
%C A212120 T(n,k) is also the total number of divisors of all positive integers <= n on the edges of k-th triangle in the diagram of divisors (see link section). See also A212119.
%H A212120 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/poldiv04.jpg">Diagram of divisors, figure 1</a>, <a href="http://www.polprimos.com/imagenespub/poldiv05.jpg">figure 2</a>.
%F A212120 T(n,k) = Sum_{j=1..n} A212119(j,k).
%e A212120 Written as an irregular triangle the sequence begins:
%e A212120 1;
%e A212120 3;
%e A212120 5;
%e A212120 7,   1;
%e A212120 9,   1;
%e A212120 11,  3;
%e A212120 13,  3;
%e A212120 15,  5;
%e A212120 17,  5,  1;
%e A212120 19,  7,  1;
%e A212120 21,  7,  1;
%e A212120 23,  9,  3;
%e A212120 25,  9,  3;
%e A212120 27, 11,  3;
%e A212120 29, 11,  5;
%e A212120 31, 13,  5,  1;
%e A212120 33, 13,  5,  1;
%e A212120 35, 15,  7,  1;
%e A212120 37, 15,  7,  1;
%e A212120 39, 17,  7,  3;
%e A212120 41, 17,  9,  3;
%e A212120 43, 19,  9,  3;
%e A212120 45, 19,  9,  3;
%e A212120 47, 21, 11,  5;
%e A212120 49, 21, 11,  5,  1;
%Y A212120 Row sums give A006218, n >= 1.
%Y A212120 Columns (1-5): A005408, A109613, A130823, A129756, A130497.
%Y A212120 Cf. A000005, A027750, A010766, A147861, A163100, A212119.
%K A212120 nonn,tabf
%O A212120 1,2
%A A212120 _Omar E. Pol_, Jul 02 2012
%E A212120 Definition changed by _Franklin T. Adams-Watters_, Jul 12 2012