A130319 Numbers k for which k!!/S(k) is an integer, where S(k) is the sum of all odd numbers less than or equal to k, if k is odd, or the sum of all even numbers less than or equal to k, if k is even.

1, 2, 6, 10, 14, 16, 17, 18, 22, 26, 28, 29, 30, 34, 38, 40, 41, 42, 46, 48, 49, 50, 52, 53, 54, 58, 62, 64, 65, 66, 68, 69, 70, 74, 76, 77, 78, 82, 86, 88, 89, 90, 94, 96, 97, 98, 100, 101, 102, 106, 108, 109, 110, 112, 113, 114, 118, 122, 124, 125, 126, 128

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..10000

Example

6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.

Maple

P:=proc(n) local a, i, j, k, w; for i from 1 by 1 to n do k:=i; w:=i-2; while w>0 do k:=k*w; w:=w-2; od; j:=i; w:=i-2; while w>0 do j:=j+w; w:=w-2; od; a:=k/j; if trunc(a)=a then print(i) fi; od; end: P(100);
# second Maple program:
q:= n-> irem(doublefactorial(n), floor((n+1)^2/4))=0:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 16 2024

Mathematica

r[n_] := If[OddQ[n], Range[1, n, 2], Range[2, n, 2]]; Select[Range[100], Divisible[Times @@ (x = r[#]), Plus @@ x] &] (* Jayanta Basu, Aug 12 2013 *)
Select[Range[100], If[OddQ[#], Divisible[#!!, ((#+1)/2)^2], Divisible[#!!, (#(#+2))/4]]&] (* Harvey P. Dale, Nov 30 2016 *)

Crossrefs

Cf. A108552, A000290, A005408, A130318.

Cf. A002620, A006882.

Sequence in context: A302796 A302797 A355530 * A336338 A191256 A067368

Adjacent sequences: A130316 A130317 A130318 * A130320 A130321 A130322

Keyword

easy,nonn,changed

Author

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

Status

approved