A301977 -id:A301977 - cp's OEIS frontend

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

Rémy Sigrist, Mar 30 2018

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

			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]

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

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

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])

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.

A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 2, 1, 2, 3, 3, 1, 2, 5, 6, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 7, 8, 7, 7, 3, 4, 9, 10, 7, 8, 7, 7, 3, 4, 7, 7, 3, 4, 3, 3, 1, 2, 5, 6, 5, 6, 11, 11, 5, 6, 13, 14, 11, 12, 11, 11, 5, 6, 13, 14

Offset: 1

Rémy Sigrist, Mar 30 2018

Equivalently, a(n) is the greatest positive number k such that A301983(n, k) = k.

Apparently, the k-th record value is A089633(k), and the first term with this value has index A048678(A089633(k)).

			The 13th row of A301983 is: 1, 2, 3, 5, 6, 7, 13; all numbers in the range 1..3 appear in this row, but the number 4 is missing; hence a(13) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A048678, A089633, A261461, A301977, A301983.

a(n) = my (b=binary(n), s=Set(1)); for (i=2, #b, s = setunion(s, Set(apply(v -> 2*v+b[i], s)))); for (u=1, oo, if (!setsearch(s,u), return (u-1)))

a(n) <= A301977(n).
a(2*n) >= a(n).
a(2*n + 1) >= a(n) (with strict inequality if a(n) is even).
a(n) = 1 iff n is positive and belongs to A000225.

A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 2, 4, 8, 11, 8, 8, 11, 8, 4, 8, 20, 34, 26, 34, 51, 40, 20, 20, 40, 51, 34, 26, 34, 20, 8, 16, 48, 96, 76, 118, 186, 152, 76, 96, 208, 281, 186, 152, 208, 124, 48, 48, 124, 208, 152, 186, 281, 208, 96, 76, 152, 186, 118, 76, 96, 48, 16, 32

Offset: 1

Rémy Sigrist, Feb 02 2023

This sequence is a variant of A165418.

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  ------------------
   1     1  N/A
   2     1  {1}
   3     1  {1}
   4     2  {1, 2}
   5     3  {1, 2, 3}
   6     3  {1, 2, 3}
   7     2  {1, 3}
   8     4  {1, 2, 4}
   9     8  {1, 2, 3, 4, 5}
  10    11  {1, 2, 3, 4, 5, 6}
  11     8  {1, 2, 3, 5, 7}
  12     8  {1, 2, 3, 4, 6}
  13    11  {1, 2, 3, 5, 6, 7}
  14     8  {1, 2, 3, 6, 7}
  15     4  {1, 3, 7}
  16     8  {1, 2, 4, 8}

Rémy Sigrist, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A165418, A301983.

{ for (n=1, #a=vector(64), print1 (a[n]=if (n==1, 1, s = [1]; b = binary(n); for (k=2, #b, s = setunion(s, apply(v -> 2*v+b[k], s))); sum(k=1, #s-1, a[s[k]]);)", ")) }

a(n) = Sum_{k = 1..A301977(n-1)} a(A301983(n, k)) for any n > 1.
a(2^k) = 2^(k-1) for any k > 0.
a(2^k-1) = 2^(k-2) for any k > 1.
a(n) >= A165418(n).

cp's OEIS Frontend

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.

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A301984 a(n) is the greatest positive number k such that the binary digits of any number in the interval 1..k appear in order but not necessarily as consecutive digits in the binary representation of n.

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A360296 a(1) = 1, and for any n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

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