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%I #14 Nov 21 2013 12:48:46
%S 1,2,30,30030,6469693230,614889782588491410,
%T 40729680599249024150621323470,
%U 2566376117594999414479597815340071648394470,225319534991831177328890236228992001350685163362356544091910
%N Product of first A000217(n) = n(n+1)/2 primes.
%C a(n) is the smallest squarefree product of a prime, 2-almost prime (semiprime), 3-almost prime, ..., n-almost prime. The analogous sequence without the squarefree condition is A006125(n), n>=2: 2,8,64,1024,32768,....
%C Cumulative product of A007467. - _Franklin T. Adams-Watters_, Mar 17 2007
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%F a(n) = prod(k=1, n*(n+1)/2, prime(k)).
%F a(n) = A002110(A000217(n)). - _Franklin T. Adams-Watters_, Mar 17 2007
%F log a(n) ~ n^2 log n. [_Charles R Greathouse IV_, Jan 13 2012]
%e a(4) = 2*(3*5)*(7*11*13)*(17*19*23*29) = 6469693230, the product of the first A000217(4) = 4*5/2 = 10 primes. 6469693230 = 2*15*1001*215441, where 2 is prime, 15 is 2-almost prime, 1001 is 3-almost prime and 215441 is 4-almost prime.
%e (Of course if the prime factors are rearranged, other primes and almost primes in the same pattern give this same product.)
%t nn=10;With[{prs=Prime[Range[(nn(nn+1))/2]]},Table[Times@@Take[prs,(n(n+1))/2], {n,0,nn}]] (* _Harvey P. Dale_, Sep 13 2011 *)
%o (PARI) a(n)=my(v=primes(n*(n+1)/2));prod(i=1,#v,v[i]) \\ _Charles R Greathouse IV_, Jan 13 2012
%Y Cf. A000217 (triangular numbers), A006125 (2^{n(n-1)/2}).
%Y Cf. A002110, A007467.
%K nonn
%O 0,2
%A _Rick L. Shepherd_, Jan 11 2006
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