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

1,4

Comments

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to binary expansion of n

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)<k<=A000043(m+1).

Example

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.

Prog

(PARI)
insert1bit(n, pos) = (((n>>pos)<<(1+pos))+(1<<pos)+bitand(n, (2^pos)-1));
binwidth(n) = { my(k=0); while(n, n>>=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

Crossrefs

Cf. A000668, A014499, A108234.

Sequence in context: A293896 A066416 A292342 * A108234 A324572 A153148

Adjacent sequences: A091988 A091989 A091990 * A091992 A091993 A091994

Keyword

nonn

Author

Reinhard Zumkeller, Mar 17 2004

Status

approved