login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A237442 a(n) is the least number of 3-smooth numbers that add up to n. 8
1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
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, 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 set of terms of the partitions of 7 {{7}, {6,1}, {5,2}, {5,1,1}, {4,3}, {4,2,1}, {4,1,1,1}, {3,3,1}, {3,2,2}, {3,2,1,1}, {3,1,1,1,1}, {2,2,2,1}, {2,2,1,1,1}, {2,1,1,1,1,1}, {1,1,1,1,1,1,1}} such that all the terms are in A003586 and none are repeated is {5,2} or {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 *)

CROSSREFS

Cf. A003586, A018899, A276380, A277071.

Sequence in context: A244259 A094840 A035218 * A277070 A139355 A039736

Adjacent sequences:  A237439 A237440 A237441 * A237443 A237444 A237445

KEYWORD

nonn,easy

AUTHOR

Lei Zhou, Feb 07 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 12 09:01 EST 2019. Contains 329052 sequences. (Running on oeis4.)