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A329571 a(n) = Product_{prime p} p^floor(log_p P) with P = A329570(n) the least prime with log_p P >= valuation(n,p) for all primes p.

%I A329571 #23 Jan 17 2025 07:49:57
%S A329571 2,2,6,60,60,6,420,27720,27720,60,27720,60,360360,420,60,12252240,
%T A329571 12252240,27720,232792560,60,420,27720,5354228880,27720,2329089562800,
%U A329571 360360,2329089562800,420,2329089562800,60,72201776446800,5342931457063200,27720,12252240,420,27720,5342931457063200,232792560,360360,27720,219060189739591200,420,9419588158802421600,27720
%N A329571 a(n) = Product_{prime p} p^floor(log_p P) with P = A329570(n) the least prime with log_p P >= valuation(n,p) for all primes p.
%C A329571 Related to the inequality (54) in Ramanujan's paper about highly composite numbers (HCN) A002182, also used in A199337: This is the square root of the (not minimal) bound a(n)^2 above which all HCN are divisible by n, according to the right part of that inequality.
%C A329571 Like the highly composite numbers A002182, all terms in this sequence are a product of primorials.
%H A329571 Amiram Eldar, <a href="/A329571/b329571.txt">Table of n, a(n) for n = 1..2308</a>
%H A329571 Srinivasa Ramanujan, <a href="https://doi.org/10.1112/plms/s2_14.1.347">Highly composite numbers</a>, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915), pp 347-409. A variant of better quality with an additional footnote is available at <a href="http://ramanujan.sirinudi.org/Volumes/published/ram15.html">Ramanujan Papers</a>.
%F A329571 a(n) = lcm([1..P]) = A003418(P) = A056604(i) with P = A329570(n), i = A000720(P).
%t A329571 a[n_] := Module[{P = NextPrime[Max[Power @@@ FactorInteger[n]] - 1], p}, p = Select[Range[P], PrimeQ]; Times @@ (p^Floor[Log[p, P]])]; a[1] = 2; Array[a, 50] (* _Amiram Eldar_, Jan 17 2025 *)
%o A329571 (PARI) apply( {A329571(n)=vecprod([p^logint(n,p)|p<-primes([2,n=A329570(n)])])}, [1..44])
%Y A329571 Cf. A329570, A002182 (highly composite numbers), A199337 (number of HCN not divisible by n), A003418 (lcm(1..n)), A056604 (lcm(1..prime(n))), A025487.
%K A329571 nonn
%O A329571 1,1
%A A329571 _M. F. Hasler_, Jan 03 2020