A328266 - OEIS

%I #43 Oct 13 2023 06:50:56

%S 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,

%T 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,

%U 44,48,39,41,45,36,35,34,41,40,49,30,47,31,27,26,43

%N 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).

%C 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.

%C a(n) >= A000720(A062383(A000040(n)))+1-n. - _Robert Israel_, Oct 17 2019

%H Rémy Sigrist, <a href="/A328266/b328266.txt">Table of n, a(n) for n = 1..10000</a>

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

%e For n = 18:

%e - prime(18) = 61,

%e - prime(19) = 67,

%e - 61 AND 67 = 1,

%e - so a(18) = 1.

%p f:= proc(n) local L,M,R,j,v,i,x;

%p   L:= convert(ithprime(n),base,2);

%p   v:= 2^nops(L);

%p   M:= select(t -> L[t]=0, [$2..nops(L)]);

%p   for i from 1 do

%p     for j from 0 to 2^nops(M)-1  do

%p       R:= convert(j,base,2);

%p       x:= 1 + add(2^(M[i]-1), i=select(k -> R[k]=1, [$1..nops(R)]))+i*v;

%p       if isprime(x) then return numtheory:-pi(x)-n fi

%p   od od;

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Oct 17 2019

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

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

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

%K nonn,base,look

%O 1,1

%A _Rémy Sigrist_, Oct 16 2019