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

%I A186193 #31 Sep 03 2025 20:09:43
%S A186193 14,20,24,26,32,34,38,44,48,50,54,56,62,64,68,74,76,80,84,86,90,92,94,
%T A186193 98,104,110,114,116,118,120,122,124,128,132,134,140,142,144,146,152,
%U A186193 154,158,160,164,168,170,174,176,182,184,186,188,194,200,202,204,206
%N A186193 Numbers k such that k!! is divisible by (k+1).
%C A186193 Positions of zeros in A119688 (k!! mod (k+1)).
%C A186193 All terms are even (for k odd, k!! is odd and not divisible by even (k+1)).
%C A186193 Apparently all integers of form 8+6*j, j >= 1, are in the sequence.
%C A186193 Exactly all even numbers k >= 14 with composite k+1 are in this sequence. - _Alois P. Heinz_, Feb 15 2011
%H A186193 Zak Seidov, <a href="/A186193/b186193.txt">Table of n, a(n) for n = 1..697</a>
%H A186193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DoubleFactorial.html">Double Factorial</a>
%e A186193 14!! = 14*12*10*8*6*4*2 = 645120 = 43008*15, so 14 is in the sequence.
%e A186193 16!! = 16*14*12*10*8*6*4*2 = 10321920 is not divisible by 17, so 16 is not in the sequence.
%e A186193 20!! = 20*18*16*14*12*10*8*6*4*2 = 3715891200 = 176947200*21, so 20 is in the sequence.
%p A186193 a:= proc(n) option remember; local k;
%p A186193       if n=1 then 14
%p A186193     else for k from 2+a(n-1) by 2
%p A186193             while isprime(k+1)
%p A186193          do od; k
%p A186193       fi
%p A186193     end:
%p A186193 seq(a(n), n=1..100);  # _Alois P. Heinz_, Feb 15 2011
%t A186193 Select[Range[300], Divisible[#!!, #+1]&] (* _Jean-François Alcover_, Nov 11 2020 *)
%o A186193 (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 A186193 Cf. A006882, A119688.
%K A186193 nonn,changed
%O A186193 1,1
%A A186193 _Zak Seidov_, Feb 14 2011