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%I #29 Sep 26 2024 15:09:13
%S 4,6,14,402,472,494,7926,13060
%N Numbers n such that A099953(n) = Sum_{i=1..n-1} (2i-1)!! is prime.
%C The first 3 associated primes are 19, 1069, 8236528396549. The next 3 terms are about 3.739... * 10^990, 2.378... * 10^1196 and 1.059... * 10^1262.
%C The sequence contains primes in the partial sums of (2n-1)!!. The corresponding primes in the partial sums of (2n)!! is the finite sequence A283563. Is this sequence also finite?
%C If for some n, (2n-1)|a(n) then a(k) is divisible by (2n-1) (and thus composite) for all k > n.
%e 6 is in this sequence because 1!! + 3!! + 5!! + 7!! + 9!! = 1 + 1*3 + 1*3*5 + 1*3*5*7 + 1*3*5*7*9 = 1069 is prime. In general, n is in this sequence if 1!! + 3!! + 5!! + ... + (2n-3)!! is prime.
%t a[n_] := Sum[(2 i - 1)!!, {i, 1, n - 1}]; Select[Range[0, 500], PrimeQ[a[#]] &]
%Y Cf. A099953, A283563.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Mar 22 2017
%E a(7) from _Vaclav Kotesovec_, Mar 28 2017
%E a(8) from _Michael S. Branicky_, Sep 26 2024