|
%I #17 Mar 16 2024 21:01:17
%S 1,2,6,10,14,16,17,18,22,26,28,29,30,34,38,40,41,42,46,48,49,50,52,53,
%T 54,58,62,64,65,66,68,69,70,74,76,77,78,82,86,88,89,90,94,96,97,98,
%U 100,101,102,106,108,109,110,112,113,114,118,122,124,125,126,128
%N Numbers k for which k!!/S(k) is 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.
%H Harvey P. Dale, <a href="/A130319/b130319.txt">Table of n, a(n) for n = 0..10000</a>
%e 6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
%e 17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.
%p 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);
%p # second Maple program:
%p q:= n-> irem(doublefactorial(n), floor((n+1)^2/4))=0:
%p select(q, [$1..200])[]; # _Alois P. Heinz_, Mar 16 2024
%t 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 *)
%t Select[Range[100],If[OddQ[#],Divisible[#!!,((#+1)/2)^2],Divisible[#!!,(#(#+2))/4]]&] (* _Harvey P. Dale_, Nov 30 2016 *)
%Y Cf. A108552, A000290, A005408, A130318.
%Y Cf. A002620, A006882.
%K easy,nonn
%O 0,2
%A _Paolo P. Lava_ and _Giorgio Balzarotti_, May 23 2007
|
