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.

2, 2, 6, 60, 60, 6, 420, 27720, 27720, 60, 27720, 60, 360360, 420, 60, 12252240, 12252240, 27720, 232792560, 60, 420, 27720, 5354228880, 27720, 2329089562800, 360360, 2329089562800, 420, 2329089562800, 60, 72201776446800, 5342931457063200, 27720, 12252240, 420, 27720, 5342931457063200, 232792560, 360360, 27720, 219060189739591200, 420, 9419588158802421600, 27720

Offset: 1

Author

M. F. Hasler, Jan 03 2020

Keywords

• nonn

Comments

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.

Like the highly composite numbers A002182, all terms in this sequence are a product of primorials.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..2308
• Srinivasa Ramanujan, Highly composite numbers, 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 Ramanujan Papers.

Crossrefs

Cf. A329570, A002182 (highly composite numbers), A199337 (number of HCN not divisible by n), A003418 (lcm(1..n)), A056604 (lcm(1..prime(n))), A025487.

Programs

Mathematica

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 *)

PARI

apply( {A329571(n)=vecprod([p^logint(n,p)|p<-primes([2,n=A329570(n)])])}, [1..44])

Formula

a(n) = lcm([1..P]) = A003418(P) = A056604(i) with P = A329570(n), i = A000720(P).