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%I #32 Jan 16 2022 09:11:17
%S 3,21,273,5187,150423,5565651,239322993,12684118629,773731236369,
%T 54934917782199,4339858504793721,386247406926641169,
%U 39010988099590758069,4174175726656211113383,471681857112151855812279,61790323281691893111408549,8588854936155173142485788311
%N a(n) = Product_{k=1..n} prime(2k), where prime(k) is the k-th prime.
%C From _Jon E. Schoenfield_, Jan 12 2022, Jan 15 2022: (Start)
%C Equivalently, a(n) is the product of the first n even-indexed primes.
%C Does r = prime(2n) * A066205(n)^2 / a(n)^2 approach a limit?
%C .
%C n A066205(n) a(n) prime(2n) r
%C ---- -------------------- -------------------- ---------- ----------
%C 1 2 3 3 1.33333...
%C 10 10156396926610 54934917782199 71 2.42683...
%C 100 4.93803636802*10^255 1.00181029905*10^257 1223 2.97141...
%C 10^3 1.5455035248*10^3740 1.2239433562*10^3742 17389 2.77263...
%C 10^4 1.760119173*10^48693 5.011593123*10^48695 224737 2.77208...
%C 10^5 3.26453781*10^596726 3.25976132*10^596729 2750159 2.75822...
%C 10^6 3.2925831*10^7045939 1.1297886*10^7045943 32452843 2.75633...
%C 10^7 8.085113*10^81117222 9.413111*10^81117226 373587883 2.75612...
%C 10^8 5.34067*10^916830347 2.09048*10^916830352 4222234741 2.75575...
%C (End)
%H Harry J. Smith, <a href="/A066206/b066206.txt">Table of n, a(n) for n = 1..100</a>
%e a(3) = prime(2) * prime(4) * prime(6) = 3 * 7 * 13 = 273.
%p a:= proc(n) option remember;
%p `if`(n=0, 1, a(n-1)*ithprime(2*n))
%p end:
%p seq(a(n), n=1..17); # _Alois P. Heinz_, Jan 12 2022
%t FoldList[Times, Prime@ Range[2, 30, 2]] (* _Michael De Vlieger_, Sep 24 2017 *)
%o (PARI) { for (n=1, 100, p=1; for (k=1, n, p*=prime(2*k)); write("b066206.txt", n, " ", p) ) } \\ _Harry J. Smith_, Feb 05 2010
%o (PARI) a(n) = prod(k=1, n, prime(2*k)); \\ _Michel Marcus_, Jan 13 2022
%Y Cf. A000040, A002110, A066205.
%K nonn
%O 1,1
%A _Leroy Quet_, Dec 16 2001
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