A130319 -id:A130319 - cp's OEIS frontend

A130318 Integer values of k!!/S(k), 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, 1, 4, 128, 11520, 143360, 425425, 2064384, 619315200, 280284364800, 6801567252480, 27512370460575, 178541140377600, 152355106455552000, 167834385271436083200, 6074006324109115392000, 29734853645550994565625, 231916605102348042240000, 392866729043377583554560000

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, May 23 2007

For n >= 8, a(n) ends with 0 or 5.

			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.

Jayanta Basu, Table of n, a(n) for n = 0..100

Cf. A108552, A000290, A005408, A130319.

Cf. A002620, A006882.

f:= n-> `if`(irem(doublefactorial(n), floor((n+1)^2/4), 'r')=0, r, [][]):
map(f, [$1..50])[];  # Alois P. Heinz, Mar 16 2024

Select[Table[Times @@ (t = If[OddQ[n], Range[1, n, 2], Range[2, n, 2]])/Plus @@ t, {n, 41}], IntegerQ] (* Jayanta Basu, Aug 12 2013 *)

Integers of the form k!!/((k+1)/2)^2, for k odd and k!!/(k*(k+2)/4) for k even. [corrected by Jon E. Schoenfield, Mar 16 2024]

A130333 Numbers k >= 1 for which k!!/Sum_{i=1..k} k is an integer.

Original entry on oeis.org

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

Paolo P. Lava and Giorgio Balzarotti, May 24 2007

			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.

Jayanta Basu, Table of n, a(n) for n = 0..1000

Cf. A130319, A130332.

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);

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 *)

More terms from Jayanta Basu, Aug 11 2013

cp's OEIS Frontend

A130318 Integer values of k!!/S(k), 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.

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