A007335 MU-numbers: next term is uniquely the product of 2 earlier terms. (Formerly M0794)

2, 3, 6, 12, 18, 24, 48, 54, 96, 162, 192, 216, 384, 486, 768, 864, 1458, 1536, 1944, 3072, 3456, 4374, 6144, 7776, 12288, 13122, 13824, 17496, 24576, 31104, 39366, 49152, 55296, 69984, 98304, 118098, 124416, 157464, 196608, 221184, 279936

Offset

1,1

Comments

All terms are 3-smooth. - Reinhard Zumkeller, Aug 13 2015

Empirically, this sequence corresponds to numbers of the form 2^v * 3^w with v = 1 or w = 1 or v and w both odd (see illustration in Links section). - Rémy Sigrist, Feb 16 2023

References

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 359.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).

Rémy Sigrist, Scatterplot of the 2-adic valuation of a(n) vs the 3-adic valuation of a(n) for n = 1..50000

Robert G. Wilson v, Note, n.d.

Formula

a(n) = A003586(A261255(n)). - Reinhard Zumkeller, Aug 13 2015

Conjecture: Sum_{n>=1} 1/a(n) = 181/144. - Amiram Eldar, Jul 31 2022

Mathematica

s={2, 3}; Do[n=Select[ Table[s[[j]] s[[k]], {j, Length@s}, {k, j+1, Length@s}] // Flatten // Sort // Split, #[[1]] > s[[-1]] && Length[#] == 1 &][[1, 1]]; AppendTo[s, n], {39}]; s  (* Jean-François Alcover, Apr 22 2011 *)
Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Times @@ # &, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {2, 3}, 39] (* Michael De Vlieger, Nov 16 2017 *)

Prog

(Haskell)
a007335 n = a007335_list !! (n-1)
a007335_list = 2 : 3 : f [3, 2] (singleton 6 1) where
   f xs m | v == 1    = y : f (y : xs) (g (map (y *) xs) m')
          | otherwise = f xs m'
          where g [] m = m
                g (z:zs) m = g zs $ insertWith (+) z 1 m
                ((y, v), m') = deleteFindMin m
-- Reinhard Zumkeller, Aug 13 2015
(Julia)
function isMU(u, n, h, i, r)
    ur = u[r]; ui = u[i]
    ur <= ui && return h
    if ur * ui > n
        r -= 1
    elseif ur * ui < n
        i += 1
    else
        h && return false
        h = true; i += 1; r -= 1
    end
    isMU(u, n, h, i, r)
end
function MUList(len)
    u = Array{Int, 1}(undef, len)
    u[1] = 2; u[2] = 3; i = 2; n = 2
    while i < len
        n += 1
        if isMU(u, n, false, 1, i)
            i += 1
            u[i] = n
        end
    end
    return u
end
MUList(41) |> println # Peter Luschny, Apr 07 2019

Crossrefs

Subsequence of A000423.

Cf. A054540, A003586, A261255.

Sequence in context: A093687 A000423 A389366 * A309311 A394760 A361693

Adjacent sequences: A007332 A007333 A007334 * A007336 A007337 A007338

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Status

approved