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

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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, Mar 17 2004

Keywords

• nonn

Comments

Insertion here means that the new 1-bit must come somewhere right of the most significant 1-bit. - Antti Karttunen, Dec 15 2017

Examples

			n = 25->'11001': A000040(16)=53->'110[1]01', therefore a(25)=1;
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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000668, A014499, A108234.

Programs

PARI

insert1bit(n,pos) = (((n>>pos)<<(1+pos))+(1<>=1;k++); k; };
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++);};
\\ Antti Karttunen, Dec 15 2017

Formula

a(2*n) = a(4*n+1) + 1.

a(A005097(n)) = 1 - A010051(A005097(n)).

a(2^k)=A061712(k); a(2^k+1)=A061712(k-1)*(1-A010051(2^k+1));

a(2^k-1) = A000043(m+1) - k for A000043(m)A000043(m+1).