This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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.)
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 *)
a(n)=my(v=primes(n*(n+1)/2));prod(i=1,#v,v[i]) \\ Charles R Greathouse IV, Jan 13 2012
Written as an irregular triangle the sequence begins:
2;
3, 2, 5;
7, 3, 11, 5, 13;
17, 7, 19, 11, 23, 13, 29;
31, 17, 37, 19, 41, 23, 43, 29, 47;
53, 31, 59, 37, 61, 41, 67, 43, 71, 47, 73;
79, 53, 83, 59, 89, 61, 97, 67, 101, 71, 103, 73, 107;
...
The triangle can be arranged as shown below so that, in every row, each odd position term is equal to the term immediately below it.
2
3 2 5
7 3 11 5 13
17 7 19 11 23 13 29
31 17 37 19 41 23 43 29 47
...
nterms=64;a=ConstantArray[0,nterms];For[n=1;p=1,n<=nterms,n++,If[a[[n]]==0,a[[n]]=Prime[p];If[(d=4p-n)<=nterms,a[[d]]=a[[n]]];p++]]; a
(* Second program, triangle rows *)
nrows=8;Table[rlen=2r-1;Permute[Prime[Range[s=1+(r-1)(r-2)/2,s+rlen-1]],Join[Range[2,rlen,2],Range[1,rlen,2]]],{r,nrows}]
Comments