1,2

The original definition left unclear whether "at least" or "exactly" n prime factors are required. Now the "at least" variant was chosen, for the other variant ("exactly"), see A069354: At least up to a(18), both criteria yield the same number, and therefore a(n) = A069354(n) - 1, since m and m+1 are always coprime. - M. F. Hasler, Jan 15 2014

10^13 < a(19) <= 69322940121435. - Giovanni Resta, Mar 24 2020

Terms a(1)-a(10) appear in Erdős and Nicolas (1978-1979). - Amiram Eldar, Jun 24 2023

Michael S. Branicky, Python program.

Paul Erdős and Jean-Louis Nicolas, Sur la fonction "nombre de facteurs premiers de n", Séminaire Delange-Pisot-Poitou, Théorie des nombres, Vol. 20, No. 2 (1978-1979), Talk no. 32, pp. 1-19. See p. 10.

a(n) = Min_{ m | A001221(m*(m+1)) >= n }.

a(n) <= A002110(n) - 1 because A001221((q-1)*q) >= n+1 for q = A002110(n).

Conjecture: a(n) = A069354(n) - 1. - Robert G. Wilson v, Feb 18 2014

For n = 9, a(9)*(a(9) + 1) = 38570*38571 = (2*5*7*19*29)*(3*13*23*43) with 9 distinct prime factors.

With[{s = Map[PrimeNu[Times @@ #] &, Partition[Range[10^6], 2, 1]]}, Array[FirstPosition[s, n_/; n>=#][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 02 2017 *)

(PARI) a(n) = my(m=1); while(omega(m*(m+1)) < n, m++); m; \\ Michel Marcus, Jul 09 2018

Cf. A001221, A002110, A006549, A054989, A083002, A232096, A232097.

Sequence in context: A049082 A158095 A227365 * A102019 A216270 A214374

Adjacent sequences: A059955 A059956 A059957 * A059959 A059960 A059961

nonn,more

Labos Elemer, Mar 02 2001

More terms from William Rex Marshall, Mar 18 2001

Offset corrected and a(15)-a(16) from Donovan Johnson, Jan 31 2009

a(17) from Donovan Johnson, Sep 15 2010

a(18) from Don Reble, Jan 15 2014

Edited by M. F. Hasler, Jan 15 2014

a(19)-a(20) from Michael S. Branicky, Feb 08 2023

a(21) from Michael S. Branicky, Feb 10 2023

a(22) from Michael S. Branicky, Feb 23 2023

approved