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A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).

%I A249120 #33 Nov 11 2024 22:23:57
%S A249120 1,4,13,5,35,20,86,65,194,175,14,415,430,56,844,970,182,1654,2075,490,
%T A249120 3133,4220,1204,30,5773,8270,2716,120,10372,15665,5810,390,18240,
%U A249120 28865,11816,1050,31449,51860,23156,2580,53292,91200,43862,5820,55,88873,157245,80822,12450,220,146095,266460,145208,25320,715
%N A249120 Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).
%C A249120 Conjecture: gives an identity for the sum of all divisors of all positive integers <= n. Alternating sum of row n equals A024916(n), i.e., Sum_{k=1..A003056(n)} (-1)^(k-1)*T(n,k) = A024916(n).
%C A249120 Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C A249120 Column 1 is A210843.
%C A249120 Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.
%C A249120 The first element of column k is A000330(k).
%C A249120 The second element of column k is A002492(k).
%e A249120 Triangle begins:
%e A249120        1;
%e A249120        4;
%e A249120       13,       5;
%e A249120       35,      20;
%e A249120       86,      65;
%e A249120      194,     175,      14;
%e A249120      415,     430,      56;
%e A249120      844,     970,     182;
%e A249120     1654,    2075,     490;
%e A249120     3133,    4220,    1204,     30;
%e A249120     5773,    8270,    2716,    120;
%e A249120    10372,   15665,    5810,    390;
%e A249120    18240,   28865,   11816,   1050;
%e A249120    31449,   51860,   23156,   2580;
%e A249120    53292,   91200,   43862,   5820,    55;
%e A249120    88873,  157245,   80822,  12450,   220;
%e A249120   146095,  266460,  145208,  25320,   715;
%e A249120   236977,  444365,  255360,  49620,  1925;
%e A249120   379746,  730475,  440286,  93990,  4730;
%e A249120   601656, 1184885,  746088, 173190, 10670;
%e A249120   943305, 1898730, 1244222, 311160, 22825,   91;
%e A249120   ...
%e A249120 For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 194, 175, 14, so the alternating row sum is 194 - 175 + 14 = 33, equaling the sum of all divisors of all positive integers <= 6.
%Y A249120 Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.
%K A249120 nonn,tabf
%O A249120 1,2
%A A249120 _Omar E. Pol_, Dec 14 2014