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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). 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 24 16:47 EST 2020. Contains 331209 sequences. (Running on oeis4.)