A356148 -id:A356148 - cp's OEIS frontend

A356149 Irregular table T(n, k), n > 0, k = 1..A356148(n), read by rows; the n-th row contains, in ascending order, the distinct positive integers whose binary expansion appears as a substring in the binary expansion of n or its complement.

1, 1, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 1, 2, 3, 6, 1, 3, 7, 1, 2, 3, 4, 7, 8, 1, 2, 3, 4, 6, 9, 1, 2, 5, 10, 1, 2, 3, 4, 5, 11, 1, 2, 3, 4, 6, 12, 1, 2, 3, 5, 6, 13, 1, 2, 3, 6, 7, 14, 1, 3, 7, 15, 1, 2, 3, 4, 7, 8, 15, 16, 1, 2, 3, 4, 6, 7, 8, 14, 17, 1, 2, 3, 4, 5, 6, 9, 13, 18

Offset: 1

Rémy Sigrist, Jul 28 2022

Leading 0's in binary expansions are ignored.

The n-th contains the n-th row of A165416.

			Table T(n, k) begins:
    1;
    1, 2;
    1, 3;
    1, 2, 3, 4;
    1, 2, 5;
    1, 2, 3, 6;
    1, 3, 7;
    1, 2, 3, 4, 7, 8;
    1, 2, 3, 4, 6, 9;
    1, 2, 5, 10;
    1, 2, 3, 4, 5, 11;
    1, 2, 3, 4, 6, 12;
    1, 2, 3, 5, 6, 13;
    1, 2, 3, 6, 7, 14;
    1, 3, 7, 15;
    1, 2, 3, 4, 7, 8, 15, 16;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..26775
Index entries for sequences related to binary expansion of n

Cf. A165416, A356148, A356150.

row(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 row(n):
    N = n.bit_length()
    c, s = ((1<> i)
            s.add((mask&c) >> i)
    return sorted(s - {0})
print([t for r in range(19) for t in row(r)]) # Michael S. Branicky, Jul 28 2022

T(n, 1) = 1.

T(n, A356148(n)) = n.

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

A356149 Irregular table T(n, k), n > 0, k = 1..A356148(n), read by rows; the n-th row contains, in ascending order, the distinct positive integers whose binary expansion appears as a substring in the binary expansion of n or its complement.

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