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

1,1

Comments

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

Links

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

Formula

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

Example

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

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A332842 A352696 A171565 * A359895 A115116 A141662

Adjacent sequences: A328263 A328264 A328265 * A328267 A328268 A328269

Keyword

nonn,base,look

Author

Rémy Sigrist, Oct 16 2019

Status

approved