Frederik P.J. Vandecasteele - cp's OEIS frontend

A380544 Numbers of the form A073138(k) XOR A038573(k).

0, 3, 5, 9, 15, 17, 27, 33, 51, 63, 65, 99, 119, 129, 195, 231, 255, 257, 387, 455, 495, 513, 771, 903, 975, 1023, 1025, 1539, 1799, 1935, 2015, 2049, 3075, 3591, 3855, 3999, 4095, 4097, 6147, 7175, 7695, 7967, 8127, 8193, 12291, 14343, 15375, 15903, 16191, 16383, 16385

Offset: 1

Frederik P.J. Vandecasteele, Jun 23 2025

The plot of the sequence has a blancmange appearance. The discontinuities in the curve are at a(n) = k^2 + 1.
The bit length of a(n+1) appears to be A000267(n)+1.
The number of 0-bits in the binary expansion of a(n+1) appears to be A082375(n).

			k = 17. A073138(17) = 24 = 11000_2. A038573(17) = 3 = 00011_2. 11000_2 XOR 00011_2 = 11011_2 = 27. 27 is a term.

Cf. A073138, A038573, A000267, A082375.

Subsequence of A006995.

s[n_, k_] := (2^n-1)*(2^(k-n)+1); Join[{0}, Table[s[n, k], {k, 2, 15}, {n, 1, Floor[k/2]}] // Flatten] (* Amiram Eldar, Jun 23 2025 *)

a8(n) = fromdigits(vecsort(binary(n), , 4), 2);
a3(n) = 2^hammingweight(n)-1;
lista(nn) = Set(vector(nn, n, bitxor(a3(n), a8(n)))); \\ Michel Marcus, Jun 23 2025

More terms from Michel Marcus, Jun 23 2025

A384314 Numbers k such that the nonzero digits in the ternary expansion k = d(1),...,d(m) satisfy d(2i+1) = d(1) and d(2i) = 3-d(1).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 10, 15, 16, 18, 20, 21, 23, 27, 29, 30, 32, 45, 47, 48, 50, 54, 55, 60, 61, 63, 64, 69, 70, 81, 82, 87, 88, 90, 91, 96, 97, 135, 136, 141, 142, 144, 145, 150, 151, 162, 164, 165, 167, 180, 182, 183, 185, 189, 191, 192, 194, 207, 209

Offset: 1

Frederik P.J. Vandecasteele, May 25 2025

The ternary expansion of the numbers in this sequence correspond to a valid linear gear train configurations with pairwise intermeshing of neighboring gears: 0 for an idler gear, 1 for a gear driven in rotational direction A, 2 for a gear driven in rotational direction B.
A gear train is valid if it has no contradictions, where a contradiction occurs if two meshed gears rotate in the same direction.
The rotation directions for the whole train are determined by the most significant ternary digit 1 or 2.
Any later driven gears must be in the same direction as the most significant when at an even distance away from there and the opposite direction when an odd distance away.

			32 is a term since its ternary expansion is
   ternary     1 0 1 2
   direction   A B A B
The direction for the 0 gear is determined by its preceding A, and the whole train has valid alternating A,B adjacent pairs.
11 is not a term because its ternary expansion 102 does not follow the pattern ABA.

A384956 Binary XOR of number of 1-bits in the binary representation of n and number of 0-bits in the binary representation of n, a(0) = 1.

Original entry on oeis.org

1, 1, 0, 2, 3, 3, 3, 3, 2, 0, 0, 2, 0, 2, 2, 4, 5, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 5, 5, 4, 6, 6, 0, 6, 0, 0, 6, 6, 0, 0, 6, 0, 6, 6, 4, 6, 0, 0, 6, 0, 6, 6, 4, 0, 6, 6, 4, 6, 4, 4, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Frederik P.J. Vandecasteele, Jun 13 2025

n=0 is taken to be a single 0 bit, and all other n are taken without leading 0 bits.

When the length of the binary representation of n is 2^k-1, then a(n) is 2^k-1.

			179 (10110011_2) has 5 (101_2) 1-bits and 3 (011_2) 0-bits. 101_2 XOR 011_2 = 110_2 = 6. a(179) = 6.

Karl-Heinz Hofmann, Log scatter plot up to 2^22

Cf. A000120, A023416, A070939, A328568.

Cf. A003987 (XOR).

a[n_] := BitXor @@ DigitCount[n, 2]; Array[a, 100, 0] (* Amiram Eldar, Jun 13 2025 *)

def A384956(n):
    if n == 0 : return 1
    return (n.bit_length() - (Ham:=n.bit_count())) ^ Ham # Karl-Heinz Hofmann, Jun 14 2025

a(n) = A000120(n) XOR A023416(n).

A383593 In the binary expansion of n, change the most significant 0 bit to 1, if there is any 0 bit.

Original entry on oeis.org

1, 1, 3, 3, 6, 7, 7, 7, 12, 13, 14, 15, 14, 15, 15, 15, 24, 25, 26, 27, 28, 29, 30, 31, 28, 29, 30, 31, 30, 31, 31, 31, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 56, 57, 58, 59, 60, 61, 62, 63, 60, 61, 62, 63, 62, 63, 63, 63, 96, 97, 98, 99, 100

Offset: 0

Frederik P.J. Vandecasteele, Jun 11 2025

n = 0 is taken to be a single 0 bit, but for all other n no leading 0 bits are used.

The plot of the sequence is fractal.

			a(25) = 29 since 25 = 11001_2 becomes 11101_2 = 29.

Michael S. Branicky, Table of n, a(n) for n = 0..16383

Cf. A000225 (fixed points), A004760 (range of values), A063250.

def a(n): return int(bin(n)[2:].replace('0', '1', 1), 2)
print([a(n) for n in range(70)]) # Michael S. Branicky, Jun 11 2025

def A383593(n): return (n if (t:=bin(n)[2:].find('0'))==-1 else n+(1<Chai Wah Wu, Jun 17 2025

a(n) = n + floor(2^(A063250(n)-1)) for n > 0. - David Radcliffe, Jun 12 2025

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User: Frederik P.J. Vandecasteele

A380544 Numbers of the form A073138(k) XOR A038573(k).

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A384314 Numbers k such that the nonzero digits in the ternary expansion k = d(1),...,d(m) satisfy d(2*i+1) = d(1) and d(2*i) = 3-d(1).

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A384956 Binary XOR of number of 1-bits in the binary representation of n and number of 0-bits in the binary representation of n, a(0) = 1.

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A383593 In the binary expansion of n, change the most significant 0 bit to 1, if there is any 0 bit.

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A384314 Numbers k such that the nonzero digits in the ternary expansion k = d(1),...,d(m) satisfy d(2i+1) = d(1) and d(2i) = 3-d(1).