A175023 Irregular table read by rows: Row n (of A175022(n) terms) contains the run-lengths in the binary representation of A175020(n), reading left to right.

1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 2, 4, 1, 4, 1, 3, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 5, 1, 5, 1, 4, 1, 1, 3, 1, 1, 1, 3, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 2, 2, 2, 3, 3, 6, 1, 6, 1, 5, 1, 1, 4, 1, 1, 1, 4, 2, 1, 3, 1, 2, 1, 3, 1, 1, 1, 1, 3, 3, 1, 2, 1, 2

Offset: 1

Author

Leroy Quet, Nov 03 2009

Keywords

• base
• nonn
• tabf

Comments

This table lists the parts of the partitions of the positive integers. Each partition is represented exactly once in this table. If n is such that 2^(m-1) <= A175020(n) <= 2^m -1, then row n of this table gives one partition of m.

Examples

			Table to start:
1
1,1
2
1,2
1,1,1
3
1,3
1,2,1
1,1,1,1
2,2
4
1,4
1,3,1
1,2,1,1
1,2,2
1,1,1,1,1
2,3
5
Note there are: 1 row that sums to 1, two rows that sum to 2, three rows that sum to 3, five rows that sum to 4, seven rows that sum to 5, etc, where 1,2,3,5,7,... are the number of unrestricted partitions of 1,2,3,4,5,...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..13055 (rows 1 <= n <= 2^11).

Crossrefs

Cf. A175020, A175022, A175024

Programs

Maple

Contribution from R. J. Mathar, Feb 27 2010: (Start)
runLSet := proc(n) option remember ; local bdg,lset,arl,p ; bdg := convert(n,base,2) ; lset := [] ; arl := -1 ; for p from 1 to nops(bdg) do if p = 1 then arl := 1 ; elif op(p,bdg) = op(p-1,bdg) then arl := arl+1 ; else if arl > 0 then lset := [arl,op(lset)] ; end if; arl := 1 ; end if; end do ; if arl > 0 then lset := [arl,op(lset)] ; end if; return lset ; end proc:
A175023 := proc(n) local thisLset,k ; thisLset := runLSet(n) ; for k from 1 to n-1 do if convert(runLSet(k),multiset) = convert(thisLset,multiset) then return ; end if; end do ; printf("%a,",thisLset) ; return ; end proc:
for n from 1 to 80 do A175023(n) ; end do; (End)

Mathematica

With[{s = Array[Sort@ Map[Length, Split@ IntegerDigits[#, 2]] &, 73]}, Map[Length /@ Split@ IntegerDigits[#, 2] &, Values[PositionIndex@ s][[All, 1]] ]] // Flatten (* Michael De Vlieger, Sep 03 2017 *)

Extensions

Terms beyond the 18th row from R. J. Mathar, Feb 27 2010