A130333 Numbers k >= 1 for which k!!/Sum_{i=1..k} k is an integer.
1, 5, 9, 13, 14, 17, 20, 21, 24, 25, 26, 29, 32, 33, 34, 37, 38, 41, 44, 45, 48, 49, 50, 53, 54, 56, 57, 61, 62, 64, 65, 68, 69, 73, 74, 76, 77, 80, 81, 84, 85, 86, 89, 90, 92, 93, 94, 97, 98, 101, 104, 105, 109, 110, 113, 114, 116, 117, 118, 120, 121, 122
Offset: 0
Examples
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.
Links
- Jayanta Basu, Table of n, a(n) for n = 0..1000
Programs
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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:=sum('w','w'=1..i); a:=k/j; if trunc(a)=a then print(i) fi; od; end: P(100); -
Mathematica
pr[n_] := Times @@ If[OddQ[n], Range[1, n, 2], Range[2, n, 2]]; Select[Range[122], Divisible[pr[#], Total[Range[#]]] &] (* Jayanta Basu, Aug 11 2013 *) Select[Range[200],Divisible[#!!,(#(#+1))/2]&] (* Harvey P. Dale, Feb 09 2015 *)
Extensions
More terms from Jayanta Basu, Aug 11 2013