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a(n) = n!! - 1.
2

%I #12 Jun 08 2021 02:33:32

%S 0,0,1,2,7,14,47,104,383,944,3839,10394,46079,135134,645119,2027024,

%T 10321919,34459424,185794559,654729074,3715891199,13749310574,

%U 81749606399,316234143224,1961990553599,7905853580624,51011754393599

%N a(n) = n!! - 1.

%C n divides a(n-1) and a(n+1) for n = {1, 2, 8, 11, 16, 19, 23, 31, 32, 43, 64, 67, 71, ...} which include all powers of 2 except 2^2 and some odd primes of the form 4k+3 belonging to A002145.

%C p^2 divides a(p-1) for odd prime p = 71.

%C p^2 divides a(p+1) for odd prime p = 23.

%C a(n) is prime for n = {3, 4, 6, 8, 16, 26, 64, 82, 90, 118, 194, 214, ...} = A007749; A007749(n) = 2*A091415(n-1) for n > 1. Corresponding primes of the form n!! - 1 are listed in A117141, cf. also A093173.

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

%F a(n) = A006882(n) - 1.

%t Table[ n!! - 1, {n,0,35} ]

%Y Cf. A006882, A002145, A117141, A093173, A007749, A091415.

%K nonn

%O 0,4

%A _Alexander Adamchuk_, Apr 18 2007