

A328899


Numbers k such that the kth repunit (10^k1)/9 sets a new record for the number of distinct prime factors.


2



1, 2, 3, 6, 12, 18, 24, 30, 42, 60, 84, 96, 120, 168, 180, 210, 240, 252, 300, 360, 420, 630, 660
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The corresponding numbers of distinct prime factors are in A095371.
a(20) > 322.
From Chai Wah Wu, Oct 30 2019: (Start)
Since A095371(19) = 40, to show that a(20) > 323 we use the fact that (10^3231)/9) is a product of 4 primes and a 271digit composite number C. We then use a computer search to show that C has no prime factor <= floor(C^(1/(414))) = 19858291. This implies that (10^3231)/9) has less than 41 distinct prime factors. Applying this same approach to 337 and 353 (the only numbers between 323 and 359 for which the complete factorization of the corresponding repunit is not known) and using the factorization of (10^3601)/9 with 44 distinct prime factors show that a(20) = 360 and A095371(20) = 44. This approach also shows that a(21) = 420 and A095371(21) = 55.
(End)
a(24) <= 840. Conjecture: a(24) = 840, a(25) = 1260, a(26) = 1680, a(27) = 1980, a(28) = 2520, a(29) = 3360, a(30) = 3780, a(31) = 3960, a(32) = 4620, a(33) = 5040, a(34) = 6300, a(35) = 7560, a(36) = 9240, a(37) = 10080.  Chai Wah Wu, Nov 01 2019


LINKS

Table of n, a(n) for n=1..23.


MATHEMATICA

r[n_] := (10^n  1)/9; L = {}; bst = 1; Do[v = PrimeNu[r[n]]; If[v > bst, bst = v; AppendTo[L, n]], {n, 60}]; L
(* or, based on the bfile of A095370: *)
w = Last /@ Cases[Import["https://oeis.org/A095370/b095370.txt", "Table"], {_Integer, _Integer}]; L={}; bst=1; Do[If[w[[j]] > bst, AppendTo[L, j]; bst = w[[j]]], {j, Length@w}]; L


CROSSREFS

Cf. A001221, A002275, A095370, A095371.
Sequence in context: A174801 A324177 A280681 * A093687 A000423 A007335
Adjacent sequences: A328896 A328897 A328898 * A328900 A328901 A328902


KEYWORD

nonn,base,hard,more


AUTHOR

Giovanni Resta, Oct 30 2019


EXTENSIONS

a(20)a(21) from Chai Wah Wu, Oct 30 2019
a(22)a(23) from Chai Wah Wu, Nov 01 2019


STATUS

approved



