A294994 -id:A294994 - cp's OEIS frontend

A303593 A294994(n) written in base 2.

10, 11, 101, 111, 1011, 1101, 11101, 10001, 10011, 10111, 11111, 101111, 100101, 101001, 101011, 111011, 111101, 110101, 1100101, 1000111, 1000011, 1001001, 1001111, 1100111, 1100001, 1101011, 1101101, 1111111, 10111111, 10010111, 10000011, 10001001, 10001011

Offset: 1

N. J. A. Sloane, May 27 2018

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Cf. A294994, A303594.

A303594 Number of bits in binary expansion of A294994(n).

Original entry on oeis.org

2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10

Offset: 1

N. J. A. Sloane, May 27 2018

nonn

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

Cf. A035100, A294994, A303593.

s = 0; v = 2; for (n=1, 101, print1 (#binary(v) ", "); s += 2^v; forprime (p=2, oo, if (!bittest(s, p) && hammingweight(bitxor(p, v))<=2, v = p; break))) \\ Rémy Sigrist, May 27 2018

A303595 Index of n-th prime in A294994, or -1 if that prime is missing.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 7, 11, 13, 14, 15, 12, 18, 16, 17, 21, 20, 22, 23, 42, 43, 25, 19, 24, 26, 27, 44, 28, 31, 32, 33, 38, 30, 37, 34, 35, 36, 40, 39, 29, 46, 47, 48, 41, 49, 51, 52, 53, 50, 45, 54, 59, 58, 60, 57, 61, 62, 56, 66, 64, 65, 63, 67, 71, 69, 70, 68, 74, 73

Offset: 1

N. J. A. Sloane, May 28 2018

nonn

Conjectured to be a permutation of the natural numbers.

N. J. A. Sloane, Table of n, a(n) for n = 1..5000

Cf. A294994.

A297929 Lexicographically earliest sequence of distinct prime numbers such that for any n > 1, a(n) is at Hamming distance one from some previous term.

Original entry on oeis.org

2, 3, 7, 5, 11, 13, 19, 17, 23, 29, 31, 37, 43, 41, 47, 53, 59, 61, 67, 71, 79, 83, 101, 97, 103, 107, 109, 113, 131, 139, 137, 151, 149, 157, 163, 167, 179, 181, 199, 197, 193, 211, 227, 229, 241, 263, 269, 271, 293, 317, 313, 281, 283, 347, 331, 353, 359

Offset: 1

Rémy Sigrist, Jan 08 2018

For any n > 1, a(n) = a(m) XOR 2^k for some m < n and k >= 0 (where XOR denotes the bitwise XOR operator).
This sequence was inspired by A294994.
Let define the binary relation R over prime numbers as follows:
- for any prime numbers p and q, p is R-related to q iff there exists a finite list of prime numbers, say (c(1), ..., c(k)), such that c(1) = p and c(k) = q and A000120(c(i) XOR c(i+1)) = 1 for i = 1..k-1,
- R is a equivalence relation,
- this sequence corresponds to the R-equivalence class of the prime number 2.
Is this sequence infinite?
Will every prime number appear?

			See illustration of the first terms in Links section.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 359854 terms (a(359855) is the first term > 2^23)
Rémy Sigrist, Illustration of the first terms
Rémy Sigrist, PARI program for A297929

Cf. A000120, A294994.

With[{nn = 56}, Nest[Function[a, Append[a, SelectFirst[Prime@ Range[3 nn/2], Function[p, And[FreeQ[a, p], AnyTrue[a, Total@ IntegerDigits[BitXor[p, #], 2] == 1 &]]]]]], {2}, nn]] (* Michael De Vlieger, Jan 14 2018 *)

PARI
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See Links section.
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cp's OEIS Frontend

A303593 A294994(n) written in base 2.

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A303594 Number of bits in binary expansion of A294994(n).

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A303595 Index of n-th prime in A294994, or -1 if that prime is missing.

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A297929 Lexicographically earliest sequence of distinct prime numbers such that for any n > 1, a(n) is at Hamming distance one from some previous term.

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Programs

Mathematica

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