A102211 -id:A102211 - cp's OEIS frontend

A102210 Number of primes that are bitwise covered by n.

0, 1, 2, 0, 1, 1, 4, 0, 0, 1, 3, 0, 2, 1, 6, 0, 1, 1, 4, 0, 2, 1, 7, 0, 1, 1, 5, 0, 4, 1, 11, 0, 0, 1, 2, 0, 2, 1, 5, 0, 1, 1, 5, 0, 4, 1, 10, 0, 1, 1, 4, 0, 4, 1, 9, 0, 2, 1, 8, 0, 8, 1, 18, 0, 0, 1, 3, 0, 1, 1, 6, 0, 1, 1, 5, 0, 3, 1, 10, 0, 1, 1, 6, 0, 2, 1, 10, 0, 3, 1, 9, 0, 6, 1, 17, 0, 1, 1, 4, 0, 4, 1

Offset: 1

Reinhard Zumkeller, Dec 30 2004

p is bitwise covered by n iff (p = (n AND p)) bitwise: A080099(n,p)=p.

			n=21->10101 -> a(21) = #{00101=5,10001=17} = 2.

Cf. A007053, A007088, A004676, A080099, A102211, A102212, A102213, A102550.

[#[p:p in PrimesUpTo(n)| p eq BitwiseAnd(n,p)] :n in [1..105] ]; // Marius A. Burtea, Jan 12 2020

a[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Array[a, 100] (* _Amiram Eldar, Jan 12 2020 *)

a(A102211(n)) = 0; a(A102212(n)) = 1; a(A102213(n)) > 1.

a(2^k-1) = A007053(k) for k > 1. - Amiram Eldar, Jan 12 2020

A102212 Numbers m with A102210(m) = 1.

Original entry on oeis.org

2, 5, 6, 10, 14, 17, 18, 22, 25, 26, 30, 34, 38, 41, 42, 46, 49, 50, 54, 58, 62, 66, 69, 70, 73, 74, 78, 81, 82, 86, 90, 94, 97, 98, 102, 106, 110, 114, 118, 122, 126, 130, 133, 134, 137, 138, 142, 145, 146, 150, 154, 158, 162, 166, 170, 174, 177, 178, 182, 186, 190

Offset: 1

Reinhard Zumkeller, Dec 30 2004

;

Amiram Eldar, Table of n, a(n) for n = 1..10000

A016825 is a subsequence.

Cf. A102210, A102211, A102213.

[k:k in [1..200]|#[p:p in PrimesUpTo(k)| p eq BitwiseAnd(k,p)] eq 1 ]; // Marius A. Burtea, Jan 12 2020

f[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Select[Range[216], f[#] == 1 &] (* _Amiram Eldar, Jan 12 2020 *)

A102210(a(n)) = 1.

A102213 Numbers m with A102210(m) > 1.

Original entry on oeis.org

3, 7, 11, 13, 15, 19, 21, 23, 27, 29, 31, 35, 37, 39, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 67, 71, 75, 77, 79, 83, 85, 87, 89, 91, 93, 95, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 131, 135, 139, 141, 143, 147, 149, 151, 153, 155, 157

Offset: 1

Reinhard Zumkeller, Dec 30 2004

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A005408.

Cf. A102210, A102211, A102212.

[k:k in [1..160]|#[p:p in PrimesUpTo(k)| p eq BitwiseAnd(k,p)] gt 1 ]; // Marius A. Burtea, Jan 12 2020

f[n_] := Count[Range[n], ?(PrimeQ[#] && BitAnd[n, #] == # &)]; Select[Range[216], f[#] > 1 &] (* _Amiram Eldar, Jan 12 2020 *)

A102210(a(n)) > 1.

A352722 Lexicographically earliest sequence of distinct positive integers such that a(1) = 2 and for any n > 0, a(n) AND a(n+1) is prime (where AND denotes the bitwise AND operator).

Original entry on oeis.org

2, 3, 6, 10, 7, 5, 13, 15, 11, 18, 14, 19, 17, 21, 25, 23, 27, 34, 22, 35, 26, 38, 66, 30, 67, 31, 29, 37, 39, 45, 41, 43, 47, 53, 55, 61, 63, 59, 70, 42, 71, 50, 74, 51, 75, 54, 130, 46, 82, 131, 58, 134, 83, 49, 81, 57, 85, 133, 69, 135, 77, 73, 79, 87, 91

Offset: 1

Rémy Sigrist, Mar 30 2022

See A352723 for the corresponding prime numbers.

Terms of A102211 clearly do not occur in this sequence. Is this a permutation of the complement of A102211 in the positive integers? - Peter Munn, Sep 02 2022

			The first terms, alongside the corresponding prime numbers, are:
  n   a(n)  a(n) AND a(n+1)
  --  ----  ---------------
   1     2                2
   2     3                2
   3     6                2
   4    10                2
   5     7                5
   6     5                5
   7    13               13
   8    15               11
   9    11                2
  10    18                2

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

Cf. A102211, A308334 (OR variant), A336817 (XOR variant), A352723.

s=0; v=2; for (n=1, 65, print1 (v", "); s+=2^v; for (w=1, oo, if (!bittest(s, w) && isprime(bitand(v,w)), v=w; break)))

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A102210 Number of primes that are bitwise covered by n.

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A102212 Numbers m with A102210(m) = 1.

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A102213 Numbers m with A102210(m) > 1.

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A352722 Lexicographically earliest sequence of distinct positive integers such that a(1) = 2 and for any n > 0, a(n) AND a(n+1) is prime (where AND denotes the bitwise AND operator).

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