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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.

%I A130319 #25 Jun 24 2025 03:42:04
%S A130319 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 A130319 54,58,62,64,65,66,68,69,70,74,76,77,78,82,86,88,89,90,94,96,97,98,
%U A130319 100,101,102,106,108,109,110,112,113,114,118,122,124,125,126,128
%N 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.
%H A130319 Harvey P. Dale, <a href="/A130319/b130319.txt">Table of n, a(n) for n = 0..10000</a>
%e A130319 6 --> 6!! = 48; 6 + 4 + 2 = 12; 48/12 = 4.
%e A130319 17 --> 17!! = 34459425; 17 + 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 81; 34459425/81 = 425425.
%p A130319 q:= n-> irem(doublefactorial(n), floor((n+1)^2/4))=0:
%p A130319 select(q, [$1..200])[];  # _Alois P. Heinz_, Mar 16 2024
%t A130319 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 A130319 Select[Range[100],If[OddQ[#],Divisible[#!!,((#+1)/2)^2],Divisible[#!!,(#(#+2))/4]]&] (* _Harvey P. Dale_, Nov 30 2016 *)
%Y A130319 Cf. A108552, A000290, A005408, A130318.
%Y A130319 Cf. A002620, A006882.
%K A130319 easy,nonn
%O A130319 0,2
%A A130319 _Paolo P. Lava_ and _Giorgio Balzarotti_, May 23 2007