A346016 Numbers k where the average number of distinct prime factors of the divisors of k sets a new record.

1, 2, 4, 6, 12, 24, 30, 60, 120, 180, 210, 420, 840, 1260, 2310, 4620, 9240, 13860, 27720, 30030, 60060, 120120, 180180, 360360, 510510, 1021020, 2042040, 3063060, 6126120, 9699690, 19399380, 38798760, 58198140, 116396280, 223092870, 446185740, 892371480, 1338557220

Offset: 1

Author

Amiram Eldar, Jul 01 2021

Keywords

• nonn

Comments

First differs from A282472 at n = 48.

The corresponding record values are 0, 1/2, 2/3, 1, 7/6, 5/4, 3/2, 5/3, 7/4, 11/6, 2, ...

All the terms are least integers of their prime signature (A025487) since f(n) = A346009(n)/A346010(n) depends only on the prime signature of n.

This sequence is infinite since f(n) is unbounded. E.g., f(A002110(n)) = n/2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..198

Crossrefs

Cf. A002110, A282472, A346009, A346010.

Subsequence of A025487.

Programs

Mathematica

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; f[p_, e_] := e/(e + 1); f[1] = 0; f[n_] := Plus @@ f @@@ FactorInteger[n]; s = {}; fmax = -1; Do[f1 = f[lps[[k]]]; If[f1 > fmax, fmax = f1; AppendTo[s, lps[[k]]]], {k, 1, Length[lps]}]; s
DeleteDuplicates[Table[{n,Mean[PrimeNu[Divisors[n]]]},{n,100000}],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Nov 24 2024 *)