A113511 - OEIS (original) (raw)

A113511

Product of first A000217(n) = n(n+1)/2 primes.

1

1, 2, 30, 30030, 6469693230, 614889782588491410, 40729680599249024150621323470, 2566376117594999414479597815340071648394470, 225319534991831177328890236228992001350685163362356544091910

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,....

FORMULA

a(n) = prod(k=1, n*(n+1)/2, prime(k)).

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 *)