A119688 -id:A119688 - cp's OEIS frontend

A186193 Numbers k such that k!! is divisible by (k+1).

14, 20, 24, 26, 32, 34, 38, 44, 48, 50, 54, 56, 62, 64, 68, 74, 76, 80, 84, 86, 90, 92, 94, 98, 104, 110, 114, 116, 118, 120, 122, 124, 128, 132, 134, 140, 142, 144, 146, 152, 154, 158, 160, 164, 168, 170, 174, 176, 182, 184, 186, 188, 194, 200, 202, 204, 206

Offset: 1

Zak Seidov, Feb 14 2011

Positions of zeros in A119688 (k!! mod (k+1)).
All terms are even (for k odd, k!! is odd and not divisible by even (k+1)).
Apparently all integers of form 8+6*j, j >= 1, are in the sequence.
Exactly all even numbers k >= 14 with composite k+1 are in this sequence. - Alois P. Heinz, Feb 15 2011

			14!! = 14*12*10*8*6*4*2 = 645120 = 43008*15, so 14 is in the sequence.
16!! = 16*14*12*10*8*6*4*2 = 10321920 is not divisible by 17, so 16 is not in the sequence.
20!! = 20*18*16*14*12*10*8*6*4*2 = 3715891200 = 176947200*21, so 20 is in the sequence.

Zak Seidov, Table of n, a(n) for n = 1..697
Eric Weisstein's World of Mathematics, Double Factorial

Cf. A006882, A119688.

DoubleFactorial:=func< n | &*[n..2 by -2] >; [ n: n in [1..250] | DoubleFactorial(n) mod (n+1) eq 0 ]; // Klaus Brockhaus, Feb 15 2011

a:= proc(n) option remember; local k;
      if n=1 then 14
    else for k from 2+a(n-1) by 2
            while isprime(k+1)
         do od; k
      fi
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 15 2011

Select[Range[300], Divisible[#!!, #+1]&] (* Jean-François Alcover, Nov 11 2020 *)

A126868 a(n) = (n+1)!! mod n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 6, 1, 6, 5, 1, 3, 8, 7, 0, 1, 13, 9, 1, 15, 0, 11, 1, 9, 0, 13, 0, 7, 17, 15, 1, 1, 0, 17, 0, 27, 6, 19, 0, 25, 9, 21, 1, 11, 0, 23, 46, 33, 0, 25, 0, 39, 30, 27, 0, 49, 0, 29, 58, 15, 50, 31, 0, 1, 0, 33, 1, 51, 0, 35, 1, 9, 27, 37, 0, 19, 0, 39, 78, 65, 0, 41, 82, 63, 0, 43

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Mar 16 2007

			n=3 (n+1)!! = 4!! = 8; (8 mod 3) = 2.
n=4 (n+1)!! = 5!! = 15; (15 mod 4) = 3.

James C. McMahon, Table of n, a(n) for n = 1..10000

Cf. A006882, A000165, A001147, A119688.

P:=proc(n) local i,j,k,w; for i from 1 by 1 to n do k:=i+1; w:=i-1; while w>0 do k:=k*w; w:=w-2; od; j:=(k mod i); print(j); od; end: P(100);

Table[Mod[(n+1)!!,n],{n,86}] (* James C. McMahon, Dec 27 2024 *)

df(n) = prod(i=0, (n-1)\2, n - 2*i );
a(n) = df(n+1) % n; \\ Michel Marcus, Aug 22 2016

cp's OEIS Frontend

A186193 Numbers k such that k!! is divisible by (k+1).

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