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).

1, 4, 13, 5, 35, 20, 86, 65, 194, 175, 14, 415, 430, 56, 844, 970, 182, 1654, 2075, 490, 3133, 4220, 1204, 30, 5773, 8270, 2716, 120, 10372, 15665, 5810, 390, 18240, 28865, 11816, 1050, 31449, 51860, 23156, 2580, 53292, 91200, 43862, 5820, 55, 88873, 157245, 80822, 12450, 220, 146095, 266460, 145208, 25320, 715

Offset: 1

Author

Omar E. Pol, Dec 14 2014

Keywords

• nonn
• tabf

Comments

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).

Row n has length A003056(n) hence the first element of column k is in row A000217(k).

Column 1 is A210843.

Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.

The first element of column k is A000330(k).

The second element of column k is A002492(k).

Examples

			Triangle begins:
       1;
       4;
      13,       5;
      35,      20;
      86,      65;
     194,     175,      14;
     415,     430,      56;
     844,     970,     182;
    1654,    2075,     490;
    3133,    4220,    1204,     30;
    5773,    8270,    2716,    120;
   10372,   15665,    5810,    390;
   18240,   28865,   11816,   1050;
   31449,   51860,   23156,   2580;
   53292,   91200,   43862,   5820,    55;
   88873,  157245,   80822,  12450,   220;
  146095,  266460,  145208,  25320,   715;
  236977,  444365,  255360,  49620,  1925;
  379746,  730475,  440286,  93990,  4730;
  601656, 1184885,  746088, 173190, 10670;
  943305, 1898730, 1244222, 311160, 22825,   91;
  ...
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.
		

Crossrefs

Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.