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