A059958 Smallest number m such that m*(m+1) has at least n distinct prime factors.

1, 2, 5, 14, 65, 209, 714, 7314, 38570, 254540, 728364, 11243154, 58524465, 812646120, 5163068910, 58720148850, 555409903685, 4339149420605, 69322940121435, 490005293940084, 5819629108725509, 76622240600506314

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..22.

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.

Formula

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

Example

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Keyword

nonn,more

Author

Labos Elemer, Mar 02 2001

Extensions

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

Status

approved