A356149 -id:A356149 - cp's OEIS frontend

A356148 a(n) is the number of positive integers whose binary expansion appears as a substring in the binary expansion of n or its complement.

1, 2, 2, 4, 3, 4, 3, 6, 6, 4, 6, 6, 6, 6, 4, 8, 9, 9, 8, 8, 5, 9, 9, 9, 8, 8, 9, 9, 9, 8, 5, 10, 12, 13, 12, 12, 12, 10, 12, 12, 12, 6, 10, 12, 12, 13, 12, 12, 12, 12, 10, 12, 10, 12, 13, 12, 12, 12, 13, 12, 12, 10, 6, 12, 15, 17, 16, 17, 17, 16, 15, 17, 15

Offset: 1

Rémy Sigrist, Jul 28 2022

Leading 0's in binary expansions are ignored.

			For n = 43:
- the binary expansion of 43 is "101011",
- it contains the positive numbers with binary expansions "1", "10", "11", "101", "1010", "1011", "10101", "101011",
- the complement of "101011" is "010100",
- it contains the positive numbers with binary expansions "1", "10", "100", "101", "1010", "10100",
- all in all, we have the following substrings: "1", "10", "11", "100", "101", "1010", "1011", "10100", "10101", "101011",
- so a(43) = 10.

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

Cf. A004277, A122953, A356149, A356150.

a(n) = { my (b=binary(n)); #setbinop((i,j) -> my (s=fromdigits(b[i..j], 2)); if (b[i], s, 2^(j-i+1)-1-s), [1..#b]) }

def a(n):
    N = n.bit_length()
    c, s = ((1<> i)
            s.add((mask&c) >> i)
    return len(s - {0})
print([a(n) for n in range(1, 74)]) # Michael S. Branicky, Jul 28 2022

a(n) >= A122953(n).
a(2^k-1) = 2^k-1 for any k >= 0.
a(2^k) = A004277(k) for any k >= 0.

A356150 a(n) is the sum of the positive integers whose binary expansion appears as a substring in the binary expansion of n or its complement.

Original entry on oeis.org

1, 3, 4, 10, 8, 12, 11, 25, 25, 18, 26, 28, 30, 33, 26, 56, 62, 61, 56, 56, 39, 63, 64, 67, 62, 66, 72, 77, 80, 78, 57, 119, 139, 143, 137, 135, 134, 119, 134, 134, 134, 81, 120, 138, 139, 147, 142, 146, 147, 148, 132, 153, 140, 157, 165, 165, 168, 174, 181

Offset: 1

Rémy Sigrist, Jul 28 2022

Leading 0's in binary expansions are ignored.

			For n = 11:
- row 11 of A356149 is 1, 2, 3, 4, 5, 11,
- so a(11) = 1 + 2 + 3 + 4 + 5 + 11 = 26.

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

Cf. A078823, A356148, A356149.

a(n) = { my (b=binary(n)); vecsum(setbinop((i,j) -> my (s=fromdigits(b[i..j],2)); if (b[i], s, 2^(j-i+1)-1-s), [1..#b])) }

def a(n):
    N = n.bit_length()
    c, s = ((1<> i)
            s.add((mask&c) >> i)
    return sum(s)
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Jul 28 2022

a(n) >= A078823(n).

a(n) Sum_{k = 1..A356148(n)} A356149(n, k).

cp's OEIS Frontend

A356148 a(n) is the number of positive integers whose binary expansion appears as a substring in the binary expansion of n or its complement.

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