%I #14 Jan 26 2022 11:45:20

%S 1,1,2,1,3,1,2,4,1,2,3,5,1,2,3,6,1,3,7,1,2,4,8,1,2,3,4,5,9,1,2,3,4,5,

%T 6,10,1,2,3,5,7,11,1,2,3,4,6,12,1,2,3,5,6,7,13,1,2,3,6,7,14,1,3,7,15,

%U 1,2,4,8,16,1,2,3,4,5,8,9,17,1,2,3,4,5,6

%N Irregular triangle read by rows T(n, k), n >= 1 and 1 <= k <= A301977(n): T(n, k) is the k-th positive number whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

%C This sequence has similarities with A119709 and A165416; there we consider consecutive digits, here not.

%H Rémy Sigrist, <a href="/A301983/b301983.txt">Rows n = 1..500 of triangle, flattened</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F T(n, 1) = 1.

%F T(n, A301977(n)) = n.

%F T(2^n, k) = 2^(k-1) for any n > 0 and k = 1..n+1.

%F T(2^n - 1, k) = 2^k - 1 for any n > 0 and k = 1..n.

%e Triangle begins:

%e 1: [1]

%e 2: [1, 2]

%e 3: [1, 3]

%e 4: [1, 2, 4]

%e 5: [1, 2, 3, 5]

%e 6: [1, 2, 3, 6]

%e 7: [1, 3, 7]

%e 8: [1, 2, 4, 8]

%e 9: [1, 2, 3, 4, 5, 9]

%e 10: [1, 2, 3, 4, 5, 6, 10]

%e 11: [1, 2, 3, 5, 7, 11]

%e 12: [1, 2, 3, 4, 6, 12]

%e 13: [1, 2, 3, 5, 6, 7, 13]

%e 14: [1, 2, 3, 6, 7, 14]

%e 15: [1, 3, 7, 15]

%e 16: [1, 2, 4, 8, 16]

%p b:= proc(n) option remember; `if`(n=0, {0},

%p map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))

%p end:

%p T:= n-> sort([(b(n) minus {0})[]])[]:

%p seq(T(n), n=1..20); # _Alois P. Heinz_, Jan 26 2022

%o (PARI) T(n,k) = my (b=binary(n), s=Set(1)); for (i=2, #b, s = setunion(s, Set(apply(v -> 2*v+b[i], s)))); return (s[k])

%Y Cf. A119709, A165416, A301977 (row length).

%K nonn,base,tabf

%O 1,3

%A _Rémy Sigrist_, Mar 30 2018