|
|
A067692
|
|
a(n) = Sum_{0 < d <= t <= n, d|n, t|n} d*t.
|
|
12
|
|
|
1, 7, 13, 35, 31, 97, 57, 155, 130, 227, 133, 497, 183, 413, 418, 651, 307, 988, 381, 1155, 762, 953, 553, 2225, 806, 1307, 1210, 2093, 871, 3242, 993, 2667, 1762, 2183, 1802, 5096, 1407, 2705, 2418, 5155, 1723, 5858
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Total area of all s X t rectangles, where the (s,t) are the pairs of divisors of n such that 1 <= s <= t. For example, when n = 4, the rectangles are 1 X 1, 1 X 2, 1 X 4, 2 X 2, 2 X 4, and 4 X 4, whose total area is a(4) = 1*1 + 1*2 + 1*4 + 2*2 + 2*4 + 4*4 = 35. - Wesley Ivan Hurt, Nov 15 2021
|
|
LINKS
|
|
|
FORMULA
|
Sum_{k=1..n} a(k) = (7/12)*zeta(3) * n^3 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023
|
|
EXAMPLE
|
a(6) = 1*1+1*2+1*3+1*6+2*2+2*3+2*6+3*3+3*6+6*6 = 1+2+3+6+4+6+12+9+18+36 = 97.
|
|
MATHEMATICA
|
Table[(DivisorSigma[1, n]^2+DivisorSigma[2, n])/2, {n, 50}] (* Harvey P. Dale, Jan 31 2015 *)
|
|
PROG
|
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|