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

1,2

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).

Links

Table of n, a(n) for n=1..55.

Example

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.

Sequence in context: A320030 A191509 A218356 * A170844 A051432 A064461

Adjacent sequences: A249117 A249118 A249119 * A249121 A249122 A249123

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 14 2014

Status

approved