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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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.

Crossrefs

Subsequence of A000423.

Cf. A054540, A003586, A261255.

Programs

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

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 *)

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