A214415 -id:A214415 - cp's OEIS frontend

A328266 a(n) is the least k > 0 such that prime(n) AND prime(n+k) <= 1 (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

2, 1, 2, 3, 2, 1, 5, 4, 4, 9, 14, 7, 6, 21, 29, 3, 27, 1, 14, 13, 11, 33, 10, 8, 7, 6, 6, 7, 3, 2, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 43, 42, 44, 48, 39, 41, 45, 36, 35, 34, 41, 40, 49, 30, 47, 31, 27, 26, 43

Offset: 1

Rémy Sigrist, Oct 16 2019

The sequence is well defined: for any n > 0, let x be such that prime(n) < 2^x; as 1 and 2^x are coprime, by Dirichlet's theorem on arithmetic progressions, there is a prime number q of the form q = 1 + k * 2^x, and prime(n) AND q <= 1, QED.

a(n) >= A000720(A062383(A000040(n)))+1-n. - Robert Israel, Oct 17 2019

			For n = 18:
- prime(18) = 61,
- prime(19) = 67,
- 61 AND 67 = 1,
- so a(18) = 1.

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

Cf. A000040, A000720, A062383, A175330, A214415.

f:= proc(n) local L,M,R,j,v,i,x;
  L:= convert(ithprime(n),base,2);
  v:= 2^nops(L);
  M:= select(t -> L[t]=0, [$2..nops(L)]);
  for i from 1 do
    for j from 0 to 2^nops(M)-1  do
      R:= convert(j,base,2);
      x:= 1 + add(2^(M[i]-1), i=select(k -> R[k]=1, [$1..nops(R)]))+i*v;
      if isprime(x) then return numtheory:-pi(x)-n fi
  od od;
end proc:
map(f, [$1..100]); # Robert Israel, Oct 17 2019

A328266[n_]:=Module[{q=n,p=Prime[n]},While[BitAnd[p,Prime[++q]]>1];q-n];Array[A328266,100] (* Paolo Xausa, Oct 13 2023 *)

{ forprime (p=2, prime(73), k=0; forprime (q=p+1, oo, k++; if (bitand(p, q)<=1, print1 (k ", "); break))) }

a(n) = 1 iff A175330(n) = 1.

A366550 Numbers k such that bitwise AND of prime(k) and prime(k+1) = 1.

Original entry on oeis.org

2, 6, 18, 54, 564, 3512, 6542, 564163, 2063689, 54400028, 5586502348, 252252704148404, 971269945245201, 3745011184713964

Offset: 1

Paolo Xausa, Oct 13 2023

Suggested by a comment by Alex Ratushnyak in A175330.

			18 is a term since prime(18) AND prime(19) = 1,
  prime(18) = 61 = binary 0111101
  prime(19) = 67 = binary 1000011
  bitwise AND    =        0000001

Positions of ones in A175330.

Cf. A007053, A214415.

A366550list[upto_]:=PrimePi[Select[2^Range[upto],BitAnd[NextPrime[#],NextPrime[#,-1]]==1&]];
A366550list[37] (* Uses formula, considering values in A214415 up to 37 *)

isok(k) = bitand(prime(k), prime(k+1)) == 1; \\ Michel Marcus, Oct 14 2023

a(n) = A007053(A214415(n-1)).

A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

Original entry on oeis.org

3, 13, 61, 251, 4093, 32749, 65521, 8388593, 33554393, 1073741789, 137438953447, 9007199254740881, 36028797018963913, 144115188075855859, 147573952589676412909, 37778931862957161709471, 75557863725914323419121, 2417851639229258349412301, 4835703278458516698824647

Offset: 1

Michel Marcus, Apr 16 2020

Cf. A175330.
Subsequence of A014234 (largest prime <= 2^n).
Cf. A214415 (exponents of corresponding powers of 2).

s = {}; p = 2; Do[q = NextPrime[p]; If[BitAnd[p, q] == 1, AppendTo[s, p]]; p = q, {10^5}]; s (* Amiram Eldar, Apr 16 2020 *)
Select[ NextPrime[ 2^Range[82], -1], BitAnd[#, NextPrime@ #] == 1 &] (* Giovanni Resta, Apr 16 2020 *)

isok(p) = isprime(p) && (bitand(p, nextprime(p+1)) == 1);

a(9)-a(10) from Amiram Eldar, Apr 16 2020

a(11)-a(19) from Giovanni Resta, Apr 16 2020

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A328266 a(n) is the least k > 0 such that prime(n) AND prime(n+k) <= 1 (where prime(n) denotes the n-th prime number and AND denotes the bitwise AND operator).

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A366550 Numbers k such that bitwise AND of prime(k) and prime(k+1) = 1.

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A334150 Primes p such that p AND q = 1, where q is the next prime after p and AND is the bitwise operation.

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Mathematica

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