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Smallest k such that k!! is a multiple of n.
3

%I #30 Apr 22 2024 14:58:27

%S 1,2,3,4,5,6,7,4,9,10,11,6,13,14,5,6,17,12,19,10,7,22,23,6,15,26,9,14,

%T 29,10,31,8,11,34,7,12,37,38,13,10,41,14,43,22,9,46,47,6,21,20,17,26,

%U 53,18,11,14,19,58,59,10,61,62,9,8,13,22,67,34,23,14,71,12,73,74,15,38,11,26

%N Smallest k such that k!! is a multiple of n.

%C a(n) <= n, with equality if n is 1, 9, a prime or twice a prime. Are those the only cases of equality? - _Robert Israel_, Apr 22 2024

%H Alois P. Heinz, <a href="/A007922/b007922.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)

%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%p V:= Vector(100): count:= 0:

%p S:= {$1..100}:

%p for k from 1 while count < 100 do

%p v:= doublefactorial(k);

%p SP:= select(t -> v mod t = 0, S);

%p count:= count + nops(SP);

%p V[convert(SP,list)]:= k;

%p S:= S minus SP;

%p od:

%p convert(V,list); # _Robert Israel_, Apr 21 2024

%t df[n_]:=Module[{k=1},While[!Divisible[k!!,n],k++];k]; Array[df,80] (* _Harvey P. Dale_, Jun 04 2013 *)

%Y Cf. A001751, A006882.

%K nonn

%O 1,2

%A R. Muller