A116624 -id:A116624 - cp's OEIS frontend

A116625 a(n) = A116624(n) XOR A116624(n+1).

3, 6, 12, 13, 15, 26, 23, 14, 24, 49, 43, 25, 51, 50, 48, 55, 54, 52, 62, 61, 60, 53, 56, 63, 91, 93, 59, 57, 58, 100, 95, 92, 101, 99, 110, 105, 111, 106, 107, 104, 197, 174, 102, 103, 98, 204, 198, 196, 202, 203, 200, 206, 207, 205, 201, 199, 216, 213, 212

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

nonn

XOR is A003987.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Bisection of A116626. Complement of A116624?.

Cf. A235262.

A116628 Positions where A116624 is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 7, 11, 26, 42, 109, 166, 373, 772, 1532, 2930, 6154, 10933, 24184, 44069, 105575, 155528, 374727, 681122, 1630332, 3077586, 6523332

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

Conjecture: all the powers of 2 occur in A116624 (implying that none occur in A116625) and they occur in ascending order, in which case A116624(a(n)) = A000079(n-1). Checked up to n=16.

Rémy Sigrist, C program

Cf. A000079, A116624, A116625.

C
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See Links section.
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a(17)-a(25) from Rémy Sigrist, Feb 14 2023

A116626 a(1)=1; a(odd n) = a(n-1) XOR a(n-2), for a(even n) we find the first i > 1 such that neither i nor (i XOR A116626(n-1)) is present in A116626(1..n-1), in which case a(n) = (i XOR A116626(n-1)).

Original entry on oeis.org

1, 3, 2, 6, 4, 12, 8, 13, 5, 15, 10, 26, 16, 23, 7, 14, 9, 24, 17, 49, 32, 43, 11, 25, 18, 51, 33, 50, 19, 48, 35, 55, 20, 54, 34, 52, 22, 62, 40, 61, 21, 60, 41, 53, 28, 56, 36, 63, 27, 91, 64, 93, 29, 59, 38, 57, 31, 58, 37, 100, 65, 95, 30, 92, 66, 101, 39, 99, 68, 110, 42

Offset: 1

Paul D. Hanna and Antti Karttunen, Feb 21 2006

nonn

This is a permutation of the natural numbers provided that A116625 is the complement of A116624. XOR is A003987.

Index entries for sequences that are permutations of the natural numbers

Cf. a(2n) = a(2n-1) XOR a(2n+1), a(2n+1) = A116624(n+1). Inverse: A116627. Bisections: A116624, A116625. Cf. A116648.

A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 5, 10, 6, 9, 7, 16, 17, 24, 14, 11, 18, 20, 13, 15, 19, 32, 36, 21, 25, 26, 22, 23, 27, 33, 37, 28, 34, 30, 29, 40, 42, 31, 64, 96, 35, 68, 38, 44, 41, 43, 39, 48, 56, 45, 65, 66, 46, 47, 67, 80, 52, 49, 57, 50, 82, 69, 97, 51, 53, 60, 54, 55, 58, 72, 73, 59, 61, 76, 70, 71, 74

Offset: 1

Thomas Scheuerle, Feb 17 2023

The lexicographically earliest permutation of positive numbers such that the nim-sum of the first k elements equals the nim-sum of k-1 elements with the element at position k either arithmetically added or subtracted.
The first occurrence of a number m >= 2^k is always m = 2^k.
All positive integers will appear in this sequence: see link section for details.

			   n    a(n)  a(n) in binary   a(1) XOR ... XOR a(n-1) in binary
------------------------------------------------------------------
   1     1          1b             0b
   2     2         10b             1b
   3     3         11b            11b
   4     4        100b             0b
   5     8       1000b           100b
   6    12       1100b          1100b
   7     5        101b             0b
...
Signed version of this sequence such that the arithmetic sum over the first k values equals the nim-sum over the first k values of the original sequence:
1, 2, -3, 4, 8, -12, 5, 10, -6, -9, 7, 16, -17, 24, -14, 11, -18, 20, -13, ...

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Scatterplot red: a(n) blue: a(1) XOR...XOR a(n-1) for n = 1...50000
Thomas Scheuerle, Proof that all positive integers will appear in this sequence
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to binary expansion of n

Cf. A000120, A116624, A269868, A330647, A360363, A375856.

function a = A360706( max_n )
    s = 0; a = []; t = [1:max_n];
    for n = 1:max_n
        k = 1;
        while (t(k) ~= bitand(s,t(k)))&&(0 ~= bitand(s,t(k)))
            k = k+1;
        end
        s = bitxor(s,t(k));
        a(n) = t(k);
        t(k) = max(t)+1; t = sort(t);
    end
end

{ m = s = 0; for (n = 1, 77, for (v = 1, oo, if (!bittest(s, v), x = bitand(m, v); if (x==0 || x==v, s += 2^v; m = bitxor(m, v); print1 (v", "); break;);););); } \\ Rémy Sigrist, Aug 31 2024

If a(m1) = 2^k and a(m2) = 2^k-1 then m1 - 2^k < 0 and m2 - (2^k-1) > 0 for k > 2.

cp's OEIS Frontend

A116625 a(n) = A116624(n) XOR A116624(n+1).

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A116628 Positions where A116624 is a power of 2.

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A116626 a(1)=1; a(odd n) = a(n-1) XOR a(n-2), for a(even n) we find the first i > 1 such that neither i nor (i XOR A116626(n-1)) is present in A116626(1..n-1), in which case a(n) = (i XOR A116626(n-1)).

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A360706 a(n) is the least positive number not yet used such that its binary representation has either all or none of its 1-bits in common with the XOR of a(1) to a(n-1).

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