OFFSET
1,2
COMMENTS
a(n) = dot_product (d_1,d_2,...,d_(tau(n)-1))*(d_2,d_3,...d_tau(n)), where d_1<d_2<...<d_tau(n), is increasing sequence of divisors of n. a(10) = dot_product (1,2,5)*(2,5,10) = 2+10+50 = 62.
a(n) = n if and only if n is prime. - Robert Israel, May 05 2014
The 4th problem of the 43rd International Mathematical Olympiad asked to prove that a(n) < n^2 and determine when a(n) is a divisor of n^2? (answer is: iff n is prime). - Bernard Schott, Nov 28 2022
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000.
The IMO Compendium, Problem 4, 43rd IMO 2002, Glasgow, UK.
FORMULA
a(p^e) = p * (p^(2e) - 1) / (p^2 -1). - Bernard Schott, Nov 28 2022
n <= a(n) < n^2 for n > 1. a(n)/n^2 can be arbitrarily close to n (proof: let n be divisible by the numbers up to k, for large enough k). a(n) > n^(3/2) for n composite. - Charles R Greathouse IV, Nov 29 2022
MAPLE
A078730:= proc(n)
local d, i;
d:= numtheory[divisors](n);
add(d[i]*d[i+1], i=1..nops(d)-1)
end proc;
seq(A078730(n), n=1..100); # Robert Israel, May 05 2014
MATHEMATICA
f[n_] := Module[{d, l, s, i}, d = Divisors[n]; l = Length[d]; s = 0; For[i = 1, i < l, i++, s = s + d[[i + 1]]*d[[i]]]; s]; Table[ f[n], {n, 1, 100}]
f[n_] := Plus @@ Times @@@ Partition[ Divisors@ n, 2, 1]; Array[f, 57] (* Robert G. Wilson v, Dec 18 2014 *)
PROG
(Haskell)
a078730 n = sum $ zipWith (*) ps $ tail ps where ps = a027750_row n
-- Reinhard Zumkeller, Dec 20 2014
(PARI) a(n) = my(d = divisors(n)); sum(k=1, #d-1, d[k]*d[k+1]); \\ Michel Marcus, Feb 15 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Dec 20 2002
STATUS
approved