A086799 -id:A086799 - cp's OEIS frontend

A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

15, 14, 13, 12, 11, 10, 9, 8, 19, 18, 17, 16, 17, 16, 16, 17, 31, 30, 29, 28, 27, 26, 25, 24, 33, 32, 32, 33, 32, 33, 34, 35, 47, 46, 45, 44, 43, 42, 41, 40, 51, 50, 49, 48, 49, 48, 48, 49, 63, 62, 61, 60, 59, 58, 57, 56, 64, 65, 66, 67, 68, 69, 70, 71, 79, 78, 77, 76, 75, 74

Offset: 0

Vladimir Shevelev, Apr 25 2012

Or (see comment in A206853) a(n)=n<+>4.

Conjecture: for n > 96, n + 1 <= a(n) <= 9n/8 + 1. - Charles R Greathouse IV, Apr 25 2012

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A205509, A205510, A205511, A205302, A205649, A205533, A122565, A206852, A206853, A206960, A207063, A209544, A209554, A007814, A086799, A182187, A182209.

hamming(n)=my(v=binary(n));sum(i=1,#v,v[i])
a(n)=my(k=n);while(hamming(bitxor(n,k++))!=4,);k \\ Charles R Greathouse IV, Apr 25 2012

Terms corrected by Charles R Greathouse IV, Apr 25 2012

A334045 Bitwise NOR of binary representation of n and n-1.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 6, 4, 4, 0, 2, 0, 0, 0, 14, 12, 12, 8, 10, 8, 8, 0, 6, 4, 4, 0, 2, 0, 0, 0, 30, 28, 28, 24, 26, 24, 24, 16, 22, 20, 20, 16, 18, 16, 16, 0, 14, 12, 12, 8, 10, 8, 8, 0, 6, 4, 4, 0, 2, 0, 0, 0, 62, 60, 60, 56, 58, 56, 56, 48, 54, 52, 52

Offset: 1

Christoph Schreier, Apr 13 2020

All terms are even.
a(1) = 0, a(2) = 0, and a(2^n + 1) = 2^n - 2 for n > 0. Are there any other cases where n - a(n) < 4? - Charles R Greathouse IV, Apr 13 2020
The answer to the above question is no. Write n as n = (2m+1)*k, i.e. k = A006519(n) is the highest power of 2 dividing n. If m = 0, a(n) = 0 and n - a(n) = n. If m > 0, then a(n) = 2v*k, where v is the 1's complement of m. Thus n-a(n) = (2(m-v)+1)*k. Since m in binary has a leading 1, m - v >= 1 and thus n - a(n) >= 3 with n - a(n) = 3 when n > 2, k = 1 and m - v = 1, i.e. m is a power of 2 and n is of the form 2^r + 1. - Chai Wah Wu, Apr 13 2020

			a(11) = 11 NOR 10 = bin 1011 NOR 1010 = bin 100 = 4.

Wikipedia, Bitwise operation

Cf. A038712 (n XOR n-1), A086799 (n OR n-1), A129760 (n AND n-1).

a:= n-> Bits[Nor](n, n-1):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 13 2020

a(n) = my(x=bitor(n-1, n)); bitneg(x, #binary(x)); \\ Michel Marcus, Apr 13 2020

def norbitwise(n):
    a = str(bin(n))[2:]
    b = str(bin(n-1))[2:]
    if len(b) < len(a):
        b = '0' + b
    c = ''
    for i in range(len(a)):
        if a[i] == b[i] and a[i] == '0':
            c += '1'
        else:
            c += '0'
    return int(c,2)

def A334045(n):
    m = n|(n-1)
    return 2**(len(bin(m))-2)-1-m # Chai Wah Wu, Apr 13 2020

A341522 a(n) = A156552(3*A005940(1+n)).

Original entry on oeis.org

2, 5, 6, 11, 10, 13, 14, 23, 18, 21, 22, 27, 26, 29, 30, 47, 34, 37, 38, 43, 42, 45, 46, 55, 50, 53, 54, 59, 58, 61, 62, 95, 66, 69, 70, 75, 74, 77, 78, 87, 82, 85, 86, 91, 90, 93, 94, 111, 98, 101, 102, 107, 106, 109, 110, 119, 114, 117, 118, 123, 122, 125, 126, 191, 130, 133, 134, 139, 138, 141, 142, 151, 146, 149

Offset: 0

Antti Karttunen, Feb 15 2021

nonn

Because the least significant 0-bit in A156552-code of any nonzero multiple of 3 is always alone (has 1-bit immediately to its left), it follows that A255068 (= A091067(n+1) - 1) gives these same terms in the ascending order.

Antti Karttunen, Table of n, a(n) for n = 0..8192
Christian Krause, LODA, an assembly language, a computational model and a tool for mining integer sequences
Index entries for sequences related to binary expansion of n

Row/column 2 of A341520. Permutation of A255068.

Cf. A005940, A007814, A156552, A086799, A014707 (characteristic function).

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };
A156552(n) = { my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res };
A341522(n) = A156552(3*A005940(1+n));

a(n) = A156552(3*A005940(1+n)).
From Antti Karttunen, Feb 23 2021: (Start)
a(n) = 1 + n + A086799(1+n). - [Conjectured by LODA-miner, and easily seen to be correct]
a(n) = 1+ 2*n + 2^A007814(1+n). - [As the above can be rewritten to this]
(End)

A381406 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) OR a(n-1) = 2^k - 1, where OR is the binary OR operation and k>=1, while the binary weight of a(n) does not equal that of a(n-1).

Original entry on oeis.org

0, 1, 3, 2, 5, 7, 4, 11, 6, 13, 10, 15, 8, 23, 9, 14, 17, 30, 19, 12, 27, 20, 31, 16, 47, 18, 29, 22, 43, 21, 46, 25, 39, 24, 55, 26, 45, 50, 61, 34, 63, 28, 51, 44, 59, 36, 91, 37, 58, 69, 62, 33, 94, 35, 60, 67, 124, 71, 56, 79, 48, 95, 32, 127, 38, 57, 70, 121, 54, 41, 86, 107, 52, 75, 117, 42, 53, 74, 119, 40, 87, 104, 151, 105, 118, 73, 126, 49, 78, 115

Offset: 0

Scott R. Shannon, Feb 22 2025

The fixed points begin 0, 1, 10, 315, 413, 415, 1551, 1559, 1797; there are likely infinitely more.

			a(4) = 5 = 101_2 as 5 is unused and a(3) = 2 = 10_2, and 101_2 OR 10_2 = 111_2 = 2^3 - 1, while the binary weights of 5 and 2 are 2 and 1 respectively.

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

Cf. A000120, A086799, A061712, A057168, A381405.

cp's OEIS Frontend

A182336 a(n) is the least m>=n, such that the Hamming distance D(n,m)=4.

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A334045 Bitwise NOR of binary representation of n and n-1.

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A341522 a(n) = A156552(3*A005940(1+n)).

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A381406 a(0) = 0; for n > 0, a(n) is the smallest unused number such that a(n) OR a(n-1) = 2^k - 1, where OR is the binary OR operation and k>=1, while the binary weight of a(n) does not equal that of a(n-1).

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