OFFSET
1,5
COMMENTS
Any number can be written as the sum of several 3-smooth numbers. The 3-smooth numbers themselves are the sum of 1 3-smooth number. Others will need more. Any number n could be written as the sum of n ones (the smallest 3-smooth number), which takes the greatest number of 3-smooth numbers. This sequence gives the minimum number of 3-smooth numbers that is needed to add up to n.
The index of first appearance of n in this sequence: 1, 5, 23, 431, ... . The first four terms are also 2-1, 3*2-1, 3*2^3-1, 3^3*2^4-1 respectively.
The smallest numbers which require 5 and 6 addends are 18431 and 3448733, respectively. - Giovanni Resta, Feb 09 2014
From Michael De Vlieger, Sep 30 2016: (Start)
Length of shortest partition of n such all elements are unique and in A003586.
Also a "canonic" representation of n in a dual-base number system (i.e., base(2,3)), as defined by the reference as having the lowest number of terms. The greedy algorithm defined in A276380 does not always render the canonic representation. a(41) = {1,4,36}, but {9,32} is the shortest possible partition of 41 such that all terms are in A003586. (End)
REFERENCES
V. Dimitrov, G. Jullien, and R. Muscedere, Multiple Number Base System Theory and Applications, 2nd ed., CRC Press, 2012, pp. 35-39.
LINKS
Lei Zhou, Table of n, a(n) for n = 1..10000
EXAMPLE
n = 23, 23 is not 3-smooth. We have 23 = 1+22 = 2+21 = ... = 11+12. None of the 11 pairs are both 3-smooth. However, we can find 23 = 1+4+18, a sum of three 3-smooth numbers. So a(23) = 3.
a(7) = 2 since the shortest partition of 7 such that all the terms are in A003586 and none are repeated is {4,3}. - Michael De Vlieger, Sep 30 2016
MATHEMATICA
SplitN[m_, mt_, a_, s_, aa_, ss_] := Block[{i, j, f, g, a0, s0, a1 = aa, s1 = ss, a2, s2, found = 0}, i = mt + 1; While[i--; (found == 0) && (i >= (m/3)), a0 = a; If[f = FactorInteger[i]; f[[Length[f], 1]] <= 3, j = m - i; s0 = s; If[g = FactorInteger[j]; g[[Length[g], 1]] <= 3, If[i >= j, a0++; AppendTo[s0, i]; If[j > 0, a0++; AppendTo[s0, j]]; If[ar > a0, ar = a0; If[a1 > a0, a1 = a0; s1 = s0]; found = 1]], a0++; AppendTo[s0, i]; If[ar > a0, {a2, s2} = SplitN[j, Min[i, j], a0, s0, a1, s1]; If[a1 > a2, a1 = a2; s1 = s2]]]]]; {a1, s1}]; (*This finds the shortest 3-smooth train in decreasing order that sums to n*) Table[ar = n; {ac, sc} = SplitN[n, n, 0, {}, n, {}]; ac, {n, 1, 87}]
a[n_] := Block[{p = Select[Range@n, FactorInteger[#][[-1, 1]] < 4 &], k = 1},
While[{} == Quiet@ IntegerPartitions[n, {k}, p, 1], k++]; k]; Array[a, 100] (* faster, Giovanni Resta, Feb 09 2014 *)
PROG
(PARI) A237442(n)={n+9>#M237442 && M237442=Vec(M237442, n+999); if(M237442[n], M237442[n], vecmax(factor(n)[, 1]) < 5, M237442[n]=1, my(m=99, k=n\2); until(m==2||!k--, m=min(A237442(k)+A237442(n-k), m)); M237442[n]=m)} \\ M. F. Hasler, Sep 14 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lei Zhou, Feb 07 2014
STATUS
approved