A130318 -id:A130318 - cp's OEIS frontend

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

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

			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.

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

Cf. A108552, A000290, A005408, A130318.

Cf. A002620, A006882.

q:= n-> irem(doublefactorial(n), floor((n+1)^2/4))=0:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 16 2024

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 *)

A130332 Integer values of n!!/sum(i=0..n,n), with n>=1.

Original entry on oeis.org

1, 1, 21, 1485, 6144, 225225, 17694720, 59520825, 6539968512, 24325703325, 145332633600, 14230536445125, 2596962041856000, 11288163762500625, 78354054748569600, 11665426077721040625, 86068915523813376000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 24 2007

After the ninth term all the other numbers end in 0 or 5.

			5!! = 5*3*1 = 15; 5+4+3+2+1 = 15; 15/15 = 1.
13!! = 13*11*9*7*5*3*1 = 135135; 13+12+11+10+9+8+7+6+5+4+3+2+1 = 91; 135135/91 = 1485.

Harvey P. Dale, Table of n, a(n) for n = 0..466

Cf. A130318, A126695, A108552.

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:=sum('w','w'=1..i); a:=k/j; if trunc(a)=a then print(a) fi; od; end: P(100);

Select[Table[n!!/((n(n+1))/2),{n,50}],IntegerQ] (* Harvey P. Dale, Jul 24 2019 *)

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica