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A059958 Smallest number m such that m*(m+1) has at least n distinct prime factors. 4
1, 2, 5, 14, 65, 209, 714, 7314, 38570, 254540, 728364, 11243154, 58524465, 812646120, 5163068910, 58720148850, 555409903685, 4339149420605 (list; graph; refs; listen; history; text; internal format)
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

LINKS

Table of n, a(n) for n=1..18.

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

STATUS

approved

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Last modified May 11 13:21 EDT 2021. Contains 343791 sequences. (Running on oeis4.)