OFFSET
1,1
COMMENTS
2*a(n) is the sum of the perimeters of all distinct rectangles that can be made whose side lengths are divisors of n.
Every divisor of n occurs tau(n) + 1 times in the coordinates of divisors of n. - David A. Corneth, Aug 25 2020
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d1|n, d2|n, d1<=d2} (d1+d2).
a(n) = sigma(n) * (tau(n) + 1). - David A. Corneth, Aug 25 2020
EXAMPLE
a(3) = 12; The divisors of 3 are {1,3}. The divisor pairs, (d1,d2), where d1 <= d2 are (1,1), (1,3) and (3,3). The sum of all the coordinates is then 1+1+1+3+3+3 = 12. So a(3) = 12.
a(4) = 28; The divisors of 4 are {1,2,4}. The divisor pairs, (d1,d2), where d1 <= d2 are (1,1), (1,2), (1,4), (2,2), (2,4) and (4,4). The sum of all the coordinates is then 1+1+1+2+1+4+2+2+2+4+4+4 = 28. So a(4) = 28.
a(5) = 18; The divisors of 5 are {1,5}. The divisor pairs, (d1,d2), where d1 <= d2 are (1,1), (1,5) and (5,5). The sum of all the coordinates is then 1+1+1+5+5+5 = 18. So a(5) = 18.
a(6) = 60; The divisors of 6 are {1,2,3,6}. The divisor pairs, (d1,d2), where d1 <= d2 are (1,1), (1,2), (1,3), (1,6), (2,2), (2,3), (2,6), (3,3), (3,6), (6,6). The sum of all the coordinates is then 1+1+1+2+1+3+1+6+2+2+2+3+2+6+3+3+3+6+6+6 = 60. So a(6) = 60.
MATHEMATICA
Table[Sum[Sum[(i + k) (1 - Ceiling[n/k] + Floor[n/k]) (1 - Ceiling[n/i] + Floor[n/i]), {i, k}], {k, n}], {n, 80}]
PROG
(PARI) a(n) = sigma(n) * (numdiv(n)+1) \\ David A. Corneth, Aug 25 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 24 2020
STATUS
approved