A164022 -id:A164022 - cp's OEIS frontend

A018800 Smallest prime that begins with n.

11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 211, 223, 23, 241, 251, 263, 271, 281, 29, 307, 31, 3203, 331, 347, 353, 367, 37, 383, 397, 401, 41, 421, 43, 443, 457, 461, 47, 487, 491, 503, 5101, 521, 53, 541, 557, 563, 571, 587, 59

Offset: 1

David W. Wilson

Conjecture: If a(n) = (n concatenated with k) then k < n. - Amarnath Murthy, May 01 2002

a(n) always exists. Proof. Suppose n is L digits long, and consider the numbers between n*10^B and n*10^B+10^C, where B > C are both large compared with L. All such numbers begin with the digits of n. Using the upper and lower bounds on pi(x) from Theorem 1 of Rosser and Schoenfeld, it follows that for sufficiently large B and C, at least one of these numbers is a prime. QED - N. J. A. Sloane, Nov 14 2014

T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Paolo P. Lava)
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), pp. 64-94.
Index entries for primes involving decimal expansion of n

Cf. A030665, A068164, A068695, A062584, A088781, A085608.
A164022 is the base-2 analog.
Cf. also A258337.
Row n=1 of A262369.

import Data.List (isPrefixOf, find); import Data.Maybe (fromJust)
a018800 n = read $ fromJust $
            find (show n `isPrefixOf`) $ map show a000040_list :: Int
-- Reinhard Zumkeller, Jul 01 2015

f:= proc(n) local x0, d,r,y;
   if isprime(n) then return(n) fi;
   for d from 1 do
     x0:= n*10^d;
     for r from 1 to 10^d-1 by 2 do
       if isprime(x0+r) then
          return(x0+r)
       fi
     od
   od
end proc:
seq(f(n),n=1..100); # Robert Israel, Dec 23 2014

Table[Function[d, FromDigits@ SelectFirst[ IntegerDigits@ Prime@ Range[10^4], Length@ # >= Length@ d && Take[#, Length@ d] == d &]][ IntegerDigits@ n], {n, 59}] (* Michael De Vlieger, May 24 2016, Version 10 *)

a(n{,base=10}) = for (l=0, oo, forprime (p=n*base^l, (n+1)*base^l-1, return (p))) \\ Rémy Sigrist, Jun 11 2017

from sympy import isprime
def a(n):
    if isprime(n): return n
    pow10 = 10
    while True:
        t, maxt = n * pow10 + 1, (n+1) * pow10
        while t < maxt:
            if isprime(t): return t
            t += 2
        pow10 *= 10
print([a(n) for n in range(1, 60)]) # Michael S. Branicky, Nov 02 2021

a(n) = prime(A085608(n)). - Michel Marcus, Oct 19 2013

A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

2, 2, 3, 3, 5, 5, 17, 7, 11, 7, 5, 19, 13, 17, 11, 13, 11, 37, 29, 19, 13, 7, 53, 23, 67, 31, 23, 17, 17, 29, 97, 41, 71, 53, 37, 19, 19, 67, 31, 101, 43, 73, 59, 41, 23, 41, 37, 71, 59, 103, 47, 79, 61, 43, 29, 11, 43, 73, 131, 61, 107, 83, 131, 97, 47, 31

Offset: 1

Alois P. Heinz, Sep 20 2015

			Square array A(n,k) begins:
:  2,  2,  3,  17,  5,  13,   7,  17, ...
:  3,  5,  7,  19, 11,  53,  29,  67, ...
:  5, 11, 13,  37, 23,  97,  31,  71, ...
:  7, 17, 29,  67, 41, 101,  59, 131, ...
: 11, 19, 31,  71, 43, 103,  61, 137, ...
: 13, 23, 53,  73, 47, 107, 113, 139, ...
: 17, 37, 59,  79, 83, 109, 127, 257, ...
: 19, 41, 61, 131, 89, 193, 227, 263, ...

Alois P. Heinz, Antidiagonals n = 1..200, flattened
Index entries for primes involving decimal expansion of n

Columns k=1-7 give: A000040, A080165, A080166, A262286, A262284, A262287, A262285.
Row n=1 gives A164022.
Main diagonal gives A262366.
Cf. A262350, A262369.

u:= (h, t)-> select(isprime, [seq(h*2^t+k, k=0..2^t-1)]):
A:= proc(n, k) local l, p;
      l:= proc() [] end; p:= proc() -1 end;
      while nops(l(k))

nmax = 14;
col[k_] := col[k] = Module[{bk = IntegerDigits[k, 2], lk, pp = {}, coe = 1}, lbk = Length[bk]; While[Length[pp] < nmax, pp = Select[Prime[Range[ coe*nmax]], Quiet@Take[IntegerDigits[#, 2], lbk] == bk&]; coe++]; pp];
A[n_, k_] := col[k][[n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 25 2021 *)

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

A108234 Minimum m such that n*2^m+k is prime, for k < 2^m. In other words, assuming you've read n out of a binary stream, a(n) is the minimum number of additional bits (appended to the least significant end of n) you must read before it is possible to obtain a prime.

Original entry on oeis.org

1, 0, 0, 2, 0, 1, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 2, 2, 1, 2, 2, 0, 1, 0, 2, 1, 2, 1, 1, 0, 3, 1, 2, 0, 3, 0, 1, 2, 3, 0, 1, 2, 1, 1, 2, 0, 1, 2, 1, 2, 2, 0, 2, 0, 2, 1, 2, 1, 4, 0, 1, 1, 2, 0, 3, 0, 1, 1, 2, 2, 1, 0, 3, 1, 2, 0, 2, 3, 1, 2, 2, 0, 1, 2, 3, 2, 2, 1, 1, 0, 1, 1, 2

Offset: 1

Mike Stay, Jun 16 2005

Somewhat related to the Riesel problem, A040081, the minimum m such that n*2^m-1 is prime.

			a(12) = 3 because 12 = 1100 in binary and 97 = 1100001 is the first prime that starts with 1100, needing 3 extra bits.

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

Cf. A040081, A091991, A164022 (smallest prime).

% and Octave.
for n=1:100;m=0;k=0;while(~isprime(n*2^m+k))k=k+1;if k==2^m k=0;m=m+1;end;end;x(n)=m;end;x

A108234(n) = { my(m=0,k=0); while(!isprime((n*2^m)+k), k=k+1; if(2^m==k, k=0; m=m+1)); m; }; \\ Antti Karttunen, Dec 16 2017, after Octave/MATLAB code

Definition clarified by Antti Karttunen, Dec 16 2017

A105888 a(n) = the smallest prime that, when written in binary, ends with the substring of 2n-1 in binary.

Original entry on oeis.org

3, 3, 5, 7, 41, 11, 13, 31, 17, 19, 53, 23, 89, 59, 29, 31, 97, 163, 37, 103, 41, 43, 109, 47, 113, 179, 53, 311, 313, 59, 61, 127, 193, 67, 197, 71, 73, 331, 461, 79, 337, 83, 853, 599, 89, 347, 349, 223, 97, 227, 101, 103, 233, 107, 109, 239, 113, 499, 373, 503, 761

Offset: 1

Leroy Quet, Aug 08 2009

			2*5-1 = 9 is 1001 in binary. Looking at the binary numbers that end with 1001: 1001 = 9 in decimal is composite; 11001 = 25 in decimal is composite. But 101001 = 41 in decimal is prime. So a(5) = 41. - Corrected by _Rémy Sigrist_, Feb 05 2020

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Cf. A164022.

isA105888 := proc(p,n) local pdgs,n21dgs ; pdgs := convert(p,base,2) ; n21dgs := convert(2*n-1,base,2) ; if nops(n21dgs) > nops(pdgs) then return false; else verify( [op(1..nops(n21dgs),n21dgs)],[op(1..nops(n21dgs),pdgs)],'sublist') ; end if; end proc: A105888 := proc(n) p := 2 ; while not isA105888(p,n) do p := nextprime(p) ; end do ; p ; end proc: seq(A105888(n),n=1..80) ; # R. J. Mathar, Dec 06 2009

pr=-16; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]

a(n) = my (m=2*n-1); forstep (p=m, oo, 2^#binary(m), if (isprime(p), return (p)))

Extended beyond a(10) by R. J. Mathar, Dec 06 2009

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A018800 Smallest prime that begins with n.

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A262365 A(n,k) is the n-th prime whose binary expansion begins with the binary expansion of k; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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

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A108234 Minimum m such that n*2^m+k is prime, for k < 2^m. In other words, assuming you've read n out of a binary stream, a(n) is the minimum number of additional bits (appended to the least significant end of n) you must read before it is possible to obtain a prime.

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A105888 a(n) = the smallest prime that, when written in binary, ends with the substring of 2n-1 in binary.

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