A078730 Sum of products of two successive divisors of n.

0, 2, 3, 10, 5, 26, 7, 42, 30, 62, 11, 116, 13, 114, 93, 170, 17, 242, 19, 280, 171, 266, 23, 476, 130, 366, 273, 528, 29, 713, 31, 682, 399, 614, 285, 1070, 37, 762, 549, 1150, 41, 1342, 43, 1264, 873, 1106, 47, 1916, 350, 1562, 921, 1752, 53, 2186, 665, 2166, 1143

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.

Index to sequences related to Olympiads.

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

Cf. A078713(n) = 2*A001157(n)-2*a(n)-n^2-1.

Cf. A027750.

Sequence in context: A175899 A328613 A064946 * A334628 A292239 A163767

Adjacent sequences: A078727 A078728 A078729 * A078731 A078732 A078733

Keyword

nonn

Author

Vladeta Jovovic, Dec 20 2002

Status

approved