A286681 -id:A286681 - cp's OEIS frontend

A286712 a(n) = SumXOR_{k=1..n} A286681(k), where SumXOR is the analog of summation under the binary XOR operation.

1, 2, 6, 4, 13, 6, 14, 8, 25, 21, 27, 28, 61, 56, 47, 32, 17, 50, 38, 52, 42, 54, 46, 56, 50, 26, 54, 28, 58, 30, 62, 96, 66, 34, 99, 72, 101, 76, 39, 104, 82, 106, 86, 44, 90, 110, 94, 112, 99, 50, 102, 52, 107, 118, 111, 112, 57, 122, 119, 60, 122, 62, 126

Offset: 1

Rémy Sigrist, May 13 2017

For any n>0, a(n) written in binary contains binary n as a substring,

For any n>0, a(n) >= n, with equality for n = 1, 2, 4, 6, 8, 17, 26, 28, 30, 34, 39, 44, 50, 52, 57, 60, 62, 64, 71, 76, 78, 83, 129, 134, 140, ...

			a(3) = A286681(1) XOR A286681(2) XOR A286681(3) = 1 XOR 3 XOR 4 = 6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Perl program for A286712

Cf. A286681.

A296099 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, SumXOR_{k=1..n} a(k) is divisible by n, where SumXOR is the analog of summation under the binary XOR operation.

Original entry on oeis.org

1, 3, 2, 4, 11, 9, 6, 8, 19, 5, 21, 7, 12, 14, 16, 30, 17, 39, 15, 45, 20, 22, 56, 46, 25, 87, 31, 33, 74, 36, 32, 62, 66, 96, 34, 72, 109, 37, 78, 54, 42, 44, 40, 86, 90, 116, 112, 94, 49, 85, 100, 52, 171, 61, 126, 60, 75, 67, 89, 65, 71, 13, 70, 124, 128

Offset: 1

Rémy Sigrist, Dec 04 2017

The partial XOR sums are given by A295907.
This sequence is a "binary" variant of A019444.
This sequence has connections with A286681; graphically, both sequences have similar fractal features; in the scatterplot of the current sequence, the rays emerging from the origin correspond to the numerous terms a(n) that are multiples of n.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 2500000 terms

Cf. A019444, A286681, A295907.

s = 0; x = 0; for (n=1, 65, for (k=1, oo, if (!bittest(s,k) && (xx=bitxor(x,k))%n==0, x = xx; s += 2^k; print1 (k ", "); break)))

cp's OEIS Frontend

A286712 a(n) = SumXOR_{k=1..n} A286681(k), where SumXOR is the analog of summation under the binary XOR operation.

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