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

1,4

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.

Links

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

Example

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,...

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 *)

Crossrefs

Cf. A175020, A175022, A175024

Sequence in context: A161223 A145672 A175024 * A128115 A091318 A198898

Adjacent sequences: A175020 A175021 A175022 * A175024 A175025 A175026

Keyword

base,nonn,tabf

Author

Leroy Quet, Nov 03 2009

Extensions

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

Status

approved