A301983 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.

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

Offset: 1

Author

Rémy Sigrist, Mar 30 2018

Keywords

• nonn
• base
• tabf

Comments

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

Examples

			Triangle begins:
   1:    [1]
   2:    [1, 2]
   3:    [1, 3]
   4:    [1, 2, 4]
   5:    [1, 2, 3, 5]
   6:    [1, 2, 3, 6]
   7:    [1, 3, 7]
   8:    [1, 2, 4, 8]
   9:    [1, 2, 3, 4, 5, 9]
  10:    [1, 2, 3, 4, 5, 6, 10]
  11:    [1, 2, 3, 5, 7, 11]
  12:    [1, 2, 3, 4, 6, 12]
  13:    [1, 2, 3, 5, 6, 7, 13]
  14:    [1, 2, 3, 6, 7, 14]
  15:    [1, 3, 7, 15]
  16:    [1, 2, 4, 8, 16]
		

Links

• Rémy Sigrist, Rows n = 1..500 of triangle, flattened
• Index entries for sequences related to binary expansion of n

Crossrefs

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

Programs

Maple

b:= proc(n) option remember; `if`(n=0, {0},
      map(x-> [x, 2*x+r][], b(iquo(n, 2, 'r'))))
    end:
T:= n-> sort([(b(n) minus {0})[]])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 26 2022

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])

Formula

T(n, 1) = 1.

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

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

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