A385417 Numbers with a record number of unordered factorizations into exponentially odd numbers (A268335).

1, 6, 24, 30, 60, 96, 120, 210, 240, 420, 480, 720, 840, 1680, 2520, 3360, 5040, 6720, 9240, 10080, 13440, 15120, 18480, 27720, 30240, 36960, 55440, 73920, 110880, 147840, 166320, 221760, 332640, 443520, 665280, 720720, 887040, 960960, 1108800, 1330560, 1441440

Offset

1,2

Comments

Indices of records of A385416.

The corresponding record values are 1, 2, 4, 5, 6, 7, 11, 15, 16, 21, 22, 25, 37, 59, 62, 83, ... (see the link for more values).

Links

Amiram Eldar, Table of n, a(n) for n = 1..86

Amiram Eldar, Table of n, a(n), A385416(a(n)) for n = 1..86.

Formula

The factorizations of the first 5 terms:

n | a(n) | A385416(a(n)) | factorizations

--+-------+---------------|------------------------------------------------------------------

1 | 1 | 1 | 1

2 | 6 | 2 | 2*3, 6

3 | 24 | 4 | 2 * 2 * 2 * 3, 2 * 2 * 6, 3 * 8, 24

4 | 30 | 5 | 2 * 3 * 5, 2 * 15, 3 * 10, 5 * 6, 30

5 | 60 | 6 | 2 * 2 * 3 * 5, 2 * 2 * 15, 2 * 3 * 10, 2 * 5 * 6, 2 * 30, 6 * 10

Mathematica

T[n_, k_] := T[n, k] = If[n == 1 || k == 1, {{}}, Flatten[Table[Map[Prepend[#, d] &, T[d, k/d]], {d, Select[Divisors[k], 1 < # <= n &]}], 1]]; f[n_] := T[n, n];
expOdd[n_] := AllTrue[FactorInteger[n][[;; , 2]], OddQ]; s[n_] := Count[f[n], _?(And @@ (expOdd /@ #) &)];
With[{lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {_, _}][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, 300}]; seq]

Crossrefs

Subsequence of A025487.

Cf. A268335, A385416.

Sequence in context: A132398 A073120 A147778 * A348714 A209452 A275302

Adjacent sequences: A385414 A385415 A385416 * A385418 A385419 A385420

Keyword

nonn

Author

Amiram Eldar, Jun 28 2025

Status

approved