%I #27 Sep 08 2022 08:45:55

%S 14,20,24,26,32,34,38,44,48,50,54,56,62,64,68,74,76,80,84,86,90,92,94,

%T 98,104,110,114,116,118,120,122,124,128,132,134,140,142,144,146,152,

%U 154,158,160,164,168,170,174,176,182,184,186,188,194,200,202,204,206

%N Numbers n such that n!! is divisible by (n+1).

%C Positions of zeros in A119688 (n!! mod (n+1)).

%C All terms are even (for n odd, n!! is odd and not divisible by even (n+1))

%C Apparently all integers of form 8+6k, k >= 1, are in the sequence.

%C Exactly all even numbers n >= 14 with composite n+1 are in this sequence. - _Alois P. Heinz_, Feb 15 2011

%H Zak Seidov, <a href="/A186193/b186193.txt">Table of n, a(n) for n = 1..697</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>

%e 14!! = 14*12*10*8*6*4*2 = 645120 = 43008*15, so 14 is in the sequence.

%e 16!! = 16*14*12*10*8*6*4*2 = 10321920 is not divisible by 17, so 16 is not in the sequence.

%e 20!! = 20*18*16*14*12*10*8*6*4*2 = 3715891200 = 176947200*21, so 20 is in the sequence.

%p a:= proc(n) option remember; local k;

%p if n=1 then 14

%p else for k from 2+a(n-1) by 2

%p while isprime(k+1)

%p do od; k

%p fi

%p end:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, Feb 15 2011

%t Select[Range[300], Divisible[#!!, #+1]&] (* _Jean-François Alcover_, Nov 11 2020 *)

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

%Y Cf. A006882, A119688.

%K nonn

%O 1,1

%A _Zak Seidov_, Feb 14 2011