A329571 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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.

FORMULA

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

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 *)

PROG

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

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.

Sequence in context: A174589 A326942 A247943 * A270358 A156529 A184712

Adjacent sequences: A329568 A329569 A329570 * A329572 A329573 A329574

KEYWORD

nonn

AUTHOR

M. F. Hasler, Jan 03 2020

STATUS

approved