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%I #18 Apr 24 2022 06:35:07
%S 0,0,0,0,1,0,0,1,1,0,0,0,2,0,0,0,0,3,3,0,0,0,1,3,3,3,1,0,0,1,2,3,3,2,
%T 1,0,0,0,1,3,4,3,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1,0,0,7,5,7,0,0,1,0,0,1,
%U 1,0,7,6,6,7,0,1,1,0,0,0,2,0,7,7,6,7,7,0,2,0,0
%N Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the intersection of R(n) and of R(k).
%C For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
%C The underlying idea is to merge in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.
%H Rémy Sigrist, <a href="/A341840/b341840.txt">Table of n, a(n) for n = 0..10010</a>
%H Rémy Sigrist, <a href="/A341840/a341840.png">Colored representation of the table for n, k < 2^10</a>
%H Rémy Sigrist, <a href="/A341840/a341840_1.png">Colored representation of the table for n, k < 2^10</a> (black pixels correspond to 0's)
%H Rémy Sigrist, <a href="/A341840/a341840.gp.txt">PARI program for A341840</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F T(n, k) = T(k, n).
%F T(m, T(n, k)) = T(T(m, n), k).
%F T(n, n) = n.
%F T(n, 0) = 0.
%F A070939(T(n, k)) <= min(A070939(n), A070939(k)).
%F A003188(T(n, k)) = A003188(n) AND A003188(k) (where AND denotes the bitwise AND operator).
%e Array T(n, k) begins:
%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e ---+--------------------------------------------------------
%e 0| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
%e 1| 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0
%e 2| 0 1 2 3 3 2 1 0 0 1 2 3 3 2 1 0
%e 3| 0 0 3 3 3 3 0 0 0 0 3 3 3 3 0 0
%e 4| 0 0 3 3 4 4 7 7 7 7 4 4 3 3 0 0
%e 5| 0 1 2 3 4 5 6 7 7 6 5 4 3 2 1 0
%e 6| 0 1 1 0 7 6 6 7 7 6 6 7 0 1 1 0
%e 7| 0 0 0 0 7 7 7 7 7 7 7 7 0 0 0 0
%e 8| 0 0 0 0 7 7 7 7 8 8 8 8 15 15 15 15
%e 9| 0 1 1 0 7 6 6 7 8 9 9 8 15 14 14 15
%e 10| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
%e 11| 0 0 3 3 4 4 7 7 8 8 11 11 12 12 15 15
%e 12| 0 0 3 3 3 3 0 0 15 15 12 12 12 12 15 15
%e 13| 0 1 2 3 3 2 1 0 15 14 13 12 12 13 14 15
%e 14| 0 1 1 0 0 1 1 0 15 14 14 15 15 14 14 15
%e 15| 0 0 0 0 0 0 0 0 15 15 15 15 15 15 15 15
%o (PARI) See Links section.
%Y Cf. A003188, A003987, A005811, A070939, A227736, A341839, A341840, A341841.
%K nonn,tabl,base
%O 0,13
%A _Rémy Sigrist_, Feb 21 2021
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