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A209082 Least power separator of the partitions of n. 1
1, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The least power separator of the partitions of n is introduced here as the least positive integer m for which the sums x(1)^m + x(2)^m + ... + x(k)^m, as {x(1),x(2),...,x(k)} ranges through the partitions of n (as in A000041), are distinct.

In the following table, d(n,h)=[number of partitions x(1),x(2),...,x(k) of n]-[number of distinct sums x(1)^m + x(2)^m + ... + x(k)^m], so that a(n) is the least h for which d(n,h)=0.

n.....d(n,1)..d(n,2)..d(n,3)..d(n,4)..d(n,5)..d(n,6)

1.....0.......0.......0.......0.......0.......0

2.....1.......0.......0.......0.......0.......0

3.....2.......0.......0.......0.......0.......0

4.....4.......0.......0.......0.......0.......0

5.....6.......0.......0.......0.......0.......0

6.....10......2.......0.......0.......0.......0

7.....14......2.......0.......0.......0.......0

8.....21......4.......2.......0.......0.......0

9.....29......9.......3.......0.......0.......0

10....41......15......6.......0.......0.......0

11....55......24......1.......0.......0.......0

12....76......38......16......0.......0.......0

13....100.....55......24......1.......0.......0

14....134.....81......39......1.......0.......0

15....175.....115.....61......2.......0.......0

16....230.....159.....91......3.......4.......0

17....296.....214.....130.....5.......7.......0

18....384.....293.....186.....7.......12......0

19....489.....384.....254.....12......20......0

20....626.....509.....349.....16......33......0...1...0

21....791.....662.....467.....27......48......0...1...0

22....1001....857.....625.....40......79......0...2...0

For 0<n<20, it appears that for d(n,q)=0 for all q>m but not for 19<n<32.  Only the first sixteen row sequences (excluding column 1) are monotonic.

LINKS

Table of n, a(n) for n=1..40.

EXAMPLE

The partitions of 6:  {6},{5,1},{4,2},{4,1,1},{3,3},{3,2,1},{3,1,1,1},{2,2,2},{2,2,1,1},{2,1,1,1,1},{1,1,1,1,1,1}.  The 11 sums x(1)+...x(k) all = 6.  The 11 sums x(1)^2+...x(k)^2 are, respectively, 36,26,20,18,18,14,12,12,10,8,6, which are not distinct.  The 11 sums x(1)^3+...x(k)^3 are, respectively, 216,126,72,66,54,36,30,24,18,12,6, and these are distinct, so that a(6)=3.

MATHEMATICA

p[n_] := IntegerPartitions[n]

p[n_, k_] := p[n]^k

s[n_, k_] := Map[Plus @@ # &, p[n, k]]

d[n_, m_] := Length[p[n]] - Length[Union[s[n, m]]]

t = Table[d[n, m], {n, 1, 40}, {m, 1, 25}]

a[n_] := First[Position[t[[n]], 0]]

Flatten[Table[a[n], {n, 1, 40}]]  (* A209082 *)

CROSSREFS

Cf. A000041.

Sequence in context: A048273 A175387 A024542 * A257684 A098424 A098428

Adjacent sequences:  A209079 A209080 A209081 * A209083 A209084 A209085

KEYWORD

nonn

AUTHOR

Clark Kimberling, Mar 07 2012

STATUS

approved

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.