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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.

%I A175023 #11 Sep 03 2017 21:30:49
%S A175023 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,
%T A175023 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,
%U A175023 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
%N 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.
%C A175023 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.
%H A175023 Michael De Vlieger, <a href="/A175023/b175023.txt">Table of n, a(n) for n = 1..13055</a> (rows 1 <= n <= 2^11).
%e A175023 Table to start:
%e A175023 1
%e A175023 1,1
%e A175023 2
%e A175023 1,2
%e A175023 1,1,1
%e A175023 3
%e A175023 1,3
%e A175023 1,2,1
%e A175023 1,1,1,1
%e A175023 2,2
%e A175023 4
%e A175023 1,4
%e A175023 1,3,1
%e A175023 1,2,1,1
%e A175023 1,2,2
%e A175023 1,1,1,1,1
%e A175023 2,3
%e A175023 5
%e A175023 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,...
%p A175023 Contribution from _R. J. Mathar_, Feb 27 2010: (Start)
%p A175023 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:
%p A175023 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:
%p A175023 for n from 1 to 80 do A175023(n) ; end do; (End)
%t A175023 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 *)
%Y A175023 Cf. A175020, A175022, A175024
%K A175023 base,nonn,tabf
%O A175023 1,4
%A A175023 _Leroy Quet_, Nov 03 2009
%E A175023 Terms beyond the 18th row from _R. J. Mathar_, Feb 27 2010