A342303 -id:A342303 - cp's OEIS frontend

A352245 a(0) = 1; for n >= 1, a(n) = the decimal value of the binary number of the index of where n first appears in the concatenation of all previous binary terms. If the binary value of n has not previously appeared then a(n) = 0.

1, 1, 0, 1, 0, 2, 1, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 8, 0, 0, 6, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 20, 0, 8, 0, 0, 0, 0, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 56, 0, 26, 1, 0, 0, 69, 0, 0, 0, 0, 0, 0, 0, 47, 20, 0, 71, 8, 84, 0, 110, 57, 0, 0, 0, 0, 0, 0, 0, 27, 6, 79, 155, 4, 2, 0, 0, 0, 0, 0, 0, 134

Offset: 0

Scott R. Shannon, Mar 09 2022

nonn

In the first 250000 terms the longest run of consecutive 0 terms is seven, the first occurrence of which starts at a(43). It is unknown if longer runs exists. See the companion sequence A352246 for the indices where a(n) = 0.

			a(1) = 1 as the binary string concatenation up to a(0) = '1', and the binary value of 1 is '1' which appears at index 1 in the string.
a(2) = 0 as the binary string concatenation up to a(1) = '11', while the binary value of 2 is '10' which does not appear in the string.
a(3) = 1 as the binary string concatenation up to a(2) = '110', and the binary value of 3 is '11' which appears at index 1 in the string.
a(5) = 2 as the binary string concatenation up to a(4) = '11010', and the binary value of 5 is '101' which appears at index 2 in the string.
a(17) = 8 as the binary string concatenation up to a(16) = '1101010100010001000', and the binary value of 17 is '10001' which appears at index 8 in the string.

Scott R. Shannon, Image of the first 100000 terms.

Cf. A352246, A342303 (from end), A351753, A341766.

from itertools import count, islice
def agen():
    b = "1"
    yield 1
    for k in count(1):
        bk = bin(k)[2:]
        idx = b.find(bk) + 1
        yield idx
        b += bin(idx)[2:]
print(list(islice(agen(), 93))) # Michael S. Branicky, Mar 18 2022

A352246 Indices k where A352245(k) = 0.

Original entry on oeis.org

2, 4, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 51, 54, 55, 57, 58, 59, 60, 61, 62, 63, 66, 70, 73, 74, 75, 76, 77, 78, 79, 86, 87, 88, 89, 90, 91, 96, 99, 101, 103, 107, 117, 118, 123, 126, 127, 140, 147, 151

Offset: 0

Scott R. Shannon, Mar 09 2022

nonn

See A352245 for further details and examples.

Scott R. Shannon, Image of the zero indices for the first 100000 terms of A352245.

Cf. A352245, A342303, A351753, A341766.

from itertools import count, islice
def agen():
    b = "1"
    for k in count(1):
        bk = bin(k)[2:]
        idx = b.find(bk) + 1
        if idx == 0: yield k
        b += bin(idx)[2:]
print(list(islice(agen(), 73))) # Michael S. Branicky, Mar 18 2022

A355715 a(0) = 0; for n > 0, a(n) is the total number of binary bits that n has in common with all previous terms.

Original entry on oeis.org

0, 0, 2, 1, 3, 2, 7, 8, 8, 9, 16, 15, 17, 17, 18, 19, 32, 35, 39, 42, 33, 36, 40, 40, 50, 50, 57, 57, 50, 49, 53, 54, 92, 91, 94, 93, 85, 87, 89, 90, 101, 105, 106, 113, 103, 109, 108, 116, 143, 146, 144, 149, 145, 151, 146, 153, 161, 169, 161, 170, 159, 169, 158, 170, 184, 192, 187, 194, 181

Offset: 0

Scott R. Shannon, Jul 15 2022

Scott R. Shannon, Table of n, a(n) for n = 0..10000

cf. A030190, A342303, A351753, A355374.

a(1) = 0 as a(0) = 0, and 0 shares no bits in common with 1.
a(2) = 2 as a(0) = 0, a(1) = 0, and 2 = 10_2 has the 0-bit in common with both previous terms.
a(3) = 1 as a(2) = 2 = 10_2 and 3 = 11_2 shares a 1-bit in common with 2.
a(6) = 7 as a(0) = 0, a(1) = 0, a(2) = a(5) = 2 = 10_2, a(4) = 3 = 11_2 and 6 = 110_2 shares four 0-bits and three 1-bits, seven bits in all, with these previous terms.

cp's OEIS Frontend

A352245 a(0) = 1; for n >= 1, a(n) = the decimal value of the binary number of the index of where n first appears in the concatenation of all previous binary terms. If the binary value of n has not previously appeared then a(n) = 0.

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A352246 Indices k where A352245(k) = 0.

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A355715 a(0) = 0; for n > 0, a(n) is the total number of binary bits that n has in common with all previous terms.

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