A119616 Second elementary symmetric function of divisors of n.

0, 2, 3, 14, 5, 47, 7, 70, 39, 97, 11, 287, 13, 163, 158, 310, 17, 533, 19, 609, 262, 343, 23, 1375, 155, 457, 390, 1043, 29, 1942, 31, 1302, 542, 733, 502, 3185, 37, 895, 718, 2945, 41, 3358, 43, 2247, 1859, 1267, 47, 5983, 399, 2697, 1142, 3017, 53, 5150, 1006

Offset

1,2

Comments

a(p)=p if p is prime and records are A002093 (highly abundant numbers). - Robert G. Wilson v, Jun 07 2006

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

Formula

a(n) = Sum_{u|n, v|n, u<v} u*v.

a(n) = (sigma_1(n)^2-sigma_2(n))/2, cf. A000203, A001157. - Vladeta Jovovic, Jun 07 2006

Sum_{k=1..n} a(k) = zeta(3) * n^3 / 4 + O(n^2*log(n)^2). - Amiram Eldar, Dec 15 2023

Example

|-------+------------------------------------------+---------------------|
|...n...|................divisors(n)...............|..s2(divisors.(n))...|
|-------+------------------------------------------+---------------------|
|...1...|....................1.....................|..........0..........|
|...2...|...................1,2....................|..........2..........|
|...3...|...................1,3....................|..........3..........|
|...4...|..................1,2,4...................|.........14..........|
|...5...|...................1,5....................|..........5..........|
|...6...|.................1,2,3,6..................|.........47..........|

Maple

a:= n-> (l-> add(add(l[i]*l[j], i=1..j-1), j=2..nops(l)))
        (sort([numtheory[divisors](n)[]])):
seq(a(n), n=1..80);  # Alois P. Heinz, Jun 25 2014

Mathematica

f[n_] := Block[{d = Divisors@n}, Sum[ d[[u]]*d[[v]], {v, 2, Length@d}, {u, v - 1}]]; Array[f, 55] (* Robert G. Wilson v *)

Prog

(PARI) a(n)=my(d=divisors(n)); sum(i=1, #d-1, sum(j=i+1, #d, d[i]*d[j])) \\ Charles R Greathouse IV, Mar 05 2013
(PARI) a(n)=(sigma(n)^2-sigma(n, 2))/2 \\ Charles R Greathouse IV, Mar 05 2013

Crossrefs

Cf. A002093, A000203, A001157, A002117, A067692.

Column k=2 of A224381.

Sequence in context: A321226 A337329 A329365 * A286968 A285818 A287759

Adjacent sequences: A119613 A119614 A119615 * A119617 A119618 A119619

Keyword

nonn,easy

Author

N. J. A. Sloane, based on email from Neven Juric (neven.juric(AT)apis-it.hr), Jun 07 2006

Status

approved