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A343835 Irregular table T(n, k), n > 0, k = 1..A069010(n), read by rows; the n-th row contains the shortest partition of n whose values belong to A023758 and can be added without carriers in binary, in descending order.

%I A343835 #9 May 02 2021 11:43:49
%S A343835 1,2,3,4,4,1,6,7,8,8,1,8,2,8,3,12,12,1,14,15,16,16,1,16,2,16,3,16,4,
%T A343835 16,4,1,16,6,16,7,24,24,1,24,2,24,3,28,28,1,30,31,32,32,1,32,2,32,3,
%U A343835 32,4,32,4,1,32,6,32,7,32,8,32,8,1,32,8,2,32,8,3
%N A343835 Irregular table T(n, k), n > 0, k = 1..A069010(n), read by rows; the n-th row contains the shortest partition of n whose values belong to A023758 and can be added without carriers in binary, in descending order.
%C A343835 In other words, the n-th row gives the numerical values of the runs of 1's in the binary expansion of n.
%H A343835 Rémy Sigrist, <a href="/A343835/b343835.txt">Table of n, a(n) for n = 1..6145</a>
%H A343835 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A343835 T(n, 1) = A342126(n).
%F A343835 T(n, A069010(n)) = A342410(n).
%F A343835 Sum_{k = 1..A069010(n)} T(n, k) = n.
%e A343835 Table begins:
%e A343835      1:   [1]
%e A343835      2:   [2]
%e A343835      3:   [3]
%e A343835      4:   [4]
%e A343835      5:   [4, 1]
%e A343835      6:   [6]
%e A343835      7:   [7]
%e A343835      8:   [8]
%e A343835      9:   [8, 1]
%e A343835     10:   [8, 2]
%e A343835     11:   [8, 3]
%e A343835     12:   [12]
%e A343835     13:   [12, 1]
%e A343835     14:   [14]
%e A343835     15:   [15]
%e A343835 Table begins in binary:
%e A343835        1:   [1]
%e A343835       10:   [10]
%e A343835       11:   [11]
%e A343835      100:   [100]
%e A343835      101:   [100, 1]
%e A343835      110:   [110]
%e A343835      111:   [111]
%e A343835     1000:   [1000]
%e A343835     1001:   [1000, 1]
%e A343835     1010:   [1000, 10]
%e A343835     1011:   [1000, 11]
%e A343835     1100:   [1100]
%e A343835     1101:   [1100, 1]
%e A343835     1110:   [1110]
%e A343835     1111:   [1111]
%o A343835 (PARI) row(n) = { my (rr=[]); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o-1)*2^z); n-=r; rr = concat(r, rr);); rr }
%Y A343835 Cf. A023758, A069010, A272011, A342126, A342410, A343757.
%K A343835 nonn,base,tabf,easy
%O A343835 1,2
%A A343835 _Rémy Sigrist_, May 01 2021