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

Author

Rémy Sigrist, Oct 16 2019

Keywords

• nonn
• base
• look

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

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

PARI

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

Formula

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