A174332 -id:A174332 - cp's OEIS frontend

A004676 Primes written in base 2.

10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 111011, 111101, 1000011, 1000111, 1001001, 1001111, 1010011, 1011001, 1100001, 1100101, 1100111, 1101011, 1101101, 1110001, 1111111, 10000011, 10001001

Offset: 1

N. J. A. Sloane

The only primes of binary weight 2 are the Fermat primes (only five are known: 11, 101, 10001, 100000001, 10000000000000001); the repunits base 2 are the Mersenne primes. - Daniel Forgues, Nov 07 2011

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 3.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Caldwell, The prime pages, The first 1000 primes.

Cf. A019434 Fermat primes (base 10), A000668 Mersenne primes (base 10).

Cf. A174332, A208238, A208241.

a004676 = a007088 . a000040  -- Reinhard Zumkeller, Aug 06 2012

FromDigits /@ IntegerDigits[ Prime[ Range[31]], 2] (* Robert G. Wilson v, Jun 30 2005 *)

a(n)=subst(Pol(binary(prime(n))), 'x, 10) \\ Charles R Greathouse IV, Mar 20 2013

a(n) = A007088(A000040(n)). - R. J. Mathar, Jun 03 2011

A208241 Smallest prime greater than n, with n as prefix in binary representation.

Original entry on oeis.org

2, 5, 7, 17, 11, 13, 29, 17, 19, 41, 23, 97, 53, 29, 31, 67, 71, 37, 79, 41, 43, 89, 47, 97, 101, 53, 109, 113, 59, 61, 127, 131, 67, 137, 71, 73, 149, 307, 79, 163, 83, 337, 173, 89, 181, 373, 191, 97, 197, 101, 103, 211, 107, 109, 223, 113, 229, 233, 239

Offset: 1

Reinhard Zumkeller, Feb 14 2013

A208238(n) <= a(n);

A174332(n) = a(A000040(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A208238, A004676, A007088, A010051, A030308, A055011 (iterated).

Cf. A164022 (greater or equal).

import Data.List (genericIndex, find, isPrefixOf)
import Data.Maybe (fromJust)
a208241 = genericIndex a208241_list
a208241_list = f nns $ filter ((== 1) . a010051' . fst) nns where
   f mms'@((m,ms):mms) pps'@((p,ps):pps) =
     if m == p then f mms' pps else q : f mms pps'
     where q = fst $ fromJust $ find ((ms `isPrefixOf`) . snd) pps'
   nns = zip [1..] $ map reverse $ tail a030308_tabf

A208241 := proc(n)
    local nbin,len,suf,sufbin,pbin,p ;
    nbin := convert(n,base,2) ;
    for len from 1 do
        for suf from 0 to 2^len-1 do
            sufbin := convert(suf,base,2) ;
            while nops(sufbin) < len do
                sufbin := [op(sufbin),0] ;
            end do:
            pbin := [op(sufbin),op(nbin)] ;
            p := add( 2^(i-1)*op(i,pbin),i=1..nops(pbin) ) ;
            if isprime(p) then
                return p ;
            end if;
        end do:
    end do:
end proc:
seq(A208241(n),n=1..50) ; # R. J. Mathar, May 06 2017

A208238 Smallest prime greater than n, containing it in binary representation.

Original entry on oeis.org

2, 2, 5, 7, 17, 11, 13, 23, 17, 19, 41, 23, 89, 29, 29, 31, 67, 71, 37, 79, 41, 43, 89, 47, 97, 89, 53, 59, 113, 59, 61, 127, 131, 67, 137, 71, 73, 101, 307, 79, 163, 83, 337, 107, 89, 109, 349, 191, 97, 113, 101, 103, 211, 107, 109, 223, 113, 229, 233, 239

Offset: 0

Reinhard Zumkeller, Feb 14 2013

nonn

a(n) <= A208241(n);

A174332(n) = a(A000040(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A208241, A004676, A007088, A010051, A030308.

import Data.List (genericIndex, find, isInfixOf)
import Data.Maybe (fromJust)
a208238 = genericIndex a208238_list
a208238_list = f nns $ filter ((== 1) . a010051' . fst) nns where
   f mms'@((m,ms):mms) pps'@((p,ps):pps) =
     if m == p then f mms' pps else q : f mms pps'
     where q = fst $ fromJust $ find ((ms `isInfixOf`) . snd) pps'
   nns = zip [0..] a030308_tabf

spgn[n_]:=Module[{idn2=IntegerDigits[n,2],p=NextPrime[n]},While[SequenceCount[ IntegerDigits[ p,2],idn2] == 0,p = NextPrime[p]];p]; Array[spgn,60,0] (* Harvey P. Dale, Dec 23 2023 *)

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A004676 Primes written in base 2.

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A208241 Smallest prime greater than n, with n as prefix in binary representation.

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A208238 Smallest prime greater than n, containing it in binary representation.

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