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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

			|-------+------------------------------------------+---------------------|
|...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..........|
		

Links

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

Crossrefs

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

Column k=2 of A224381.

Programs

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 *)

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

Formula

a(n) = Sum_{u|n, v|n, u

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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