OFFSET
0,2
COMMENTS
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,....
Cumulative product of A007467. - Franklin T. Adams-Watters, Mar 17 2007
FORMULA
a(n) = prod(k=1, n*(n+1)/2, prime(k)).
log a(n) ~ n^2 log n. [Charles R Greathouse IV, Jan 13 2012]
EXAMPLE
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.
(Of course if the prime factors are rearranged, other primes and almost primes in the same pattern give this same product.)
MATHEMATICA
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 *)
PROG
(PARI) a(n)=my(v=primes(n*(n+1)/2)); prod(i=1, #v, v[i]) \\ Charles R Greathouse IV, Jan 13 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Rick L. Shepherd, Jan 11 2006
STATUS
approved