cp's OEIS Frontend

A091991 Minimal number of 1's that must be inserted into the binary representation of n to get a prime.

%I A091991 #15 Dec 18 2017 11:55:22
%S A091991 1,0,0,2,0,1,0,1,1,2,0,2,0,1,1,2,0,1,0,1,1,2,0,2,1,1,1,4,0,1,0,2,1,2,
%T A091991 1,1,0,2,1,2,0,2,0,1,1,3,0,1,1,1,1,2,0,1,2,1,3,3,0,3,0,2,1,3,1,2,0,1,
%U A091991 1,2,0,2,0,1,1,2,1,1,0,2,1,2,0,3,1,1,2,2,0,1,2,2,2,2,1,1,0,1,1,2,0,2
%N A091991 Minimal number of 1's that must be inserted into the binary representation of n to get a prime.
%C A091991 Insertion here means that the new 1-bit must come somewhere right of the most significant 1-bit. - _Antti Karttunen_, Dec 15 2017
%H A091991 Antti Karttunen, <a href="/A091991/b091991.txt">Table of n, a(n) for n = 1..16384</a>
%H A091991 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A091991 a(2*n) = a(4*n+1) + 1.
%F A091991 a(A005097(n)) = 1 - A010051(A005097(n)).
%F A091991 a(2^k)=A061712(k); a(2^k+1)=A061712(k-1)*(1-A010051(2^k+1));
%F A091991 a(2^k-1) = A000043(m+1) - k for A000043(m)<k<=A000043(m+1).
%e A091991 n = 25->'11001': A000040(16)=53->'110[1]01', therefore a(25)=1;
%e A091991 a(255)=a(2^8-1)=5, as 2^(8+5)-1=8191 is a Mersenne prime and 2^(8+i)-1 is not prime for i<5.
%o A091991 (PARI)
%o A091991 insert1bit(n,pos) = (((n>>pos)<<(1+pos))+(1<<pos)+bitand(n,(2^pos)-1));
%o A091991 binwidth(n) = { my(k=0); while(n,n>>=1;k++); k; };
%o A091991 A091991(n) = { if(1==n,return(1)); if(isprime(n),return(0)); if(!(n%2),return(1+A091991(1+n+n))); my(k,nexttries,prevtries = Set([n]), w = binwidth(n)-1); for(b=1,oo,nexttries = Set([]); for(t=1,length(prevtries), h = prevtries[t]; for(i=1,w,if(isprime(k=insert1bit(h,i)),return(b),nexttries = setunion(Set([k]),nexttries)))); prevtries = nexttries; w++);};
%o A091991 \\ _Antti Karttunen_, Dec 15 2017
%Y A091991 Cf. A000668, A014499, A108234.
%K A091991 nonn
%O A091991 1,4
%A A091991 _Reinhard Zumkeller_, Mar 17 2004