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 A091991 Minimal number of 1's that must be inserted into the binary representation of n to get a prime. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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 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)'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<>=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

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Last modified August 1 17:43 EDT 2021. Contains 346402 sequences. (Running on oeis4.)