A258190 -id:A258190 - cp's OEIS frontend

A338715 Smallest prime ending with decimal expansion of n, for n relatively prime to 10.

11, 3, 7, 19, 11, 13, 17, 19, 421, 23, 127, 29, 31, 233, 37, 139, 41, 43, 47, 149, 151, 53, 157, 59, 61, 163, 67, 269, 71, 73, 277, 79, 181, 83, 487, 89, 191, 193, 97, 199, 101, 103, 107, 109, 2111, 113, 1117, 3119, 3121, 1123, 127, 1129, 131, 4133, 137, 139, 2141, 2143, 5147, 149, 151, 1153, 157

Offset: 1

N. J. A. Sloane, Nov 11 2020

a(n) exists by Dirichlet's theorem.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for primes involving decimal expansion of n

Cf. A045572, A105888 (base 2 equivalent), A258190.

See A245193, A337834, A338716 for other versions.

N:= 100: # for a(1) to a(N)
V:= Vector(N):
count:= 0:
for n from 1 while count < N do
  if igcd(n,10)=1 then
    count:= count+1;
    d:= ilog10(n)+1;
    for x from n by 10^d do
      if isprime(x) then V[count]:= x; break fi
    od
  fi
od:
convert(V,list); # Robert Israel, Nov 11 2020

from sympy import isprime
def a(n):
    ending = 2*n - 1 + (n+1)//4 * 2 # A045572
    i, pow10 = ending, 10**len(str(ending))
    while not isprime(i): i += pow10
    return i
print([a(n) for n in range(1, 64)]) # Michael S. Branicky, Nov 03 2021

More terms from Robert Israel, Nov 11 2020

A258337 Smallest prime starting with n that does not appear earlier.

Original entry on oeis.org

11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 113, 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, 419, 421, 43, 443, 457, 461, 47, 487, 491, 503, 5101, 521, 53, 541, 557, 563, 571, 587, 59, 601, 613, 6203, 631, 641, 653, 661, 67, 683, 691, 701

Offset: 1

Vladimir Shevelev, May 27 2015

According to a comment in A018800, every term exists. So the sequence is a permutation of the primes. Indeed, every prime p appears in the sequence either as the p-th term or earlier.

Peter J. C. Moses, Table of n, a(n) for n = 1..5000
Index entries for primes involving decimal expansion of n

Cf. A000040, A018800, A030665, A062584, A258190, A258339.

import Data.List (isPrefixOf, delete)
a258337 n = a258337_list !! (n-1)
a258337_list = f 1 $ map show a000040_list where
   f x pss = g pss where
     g (qs:qss) = if show x `isPrefixOf` qs
                     then (read qs :: Int) : f (x + 1) (delete qs pss)
                     else g qss
-- Reinhard Zumkeller, Jul 01 2015

N:= 100: # to get a(1) to a(N)
for n from 1 to N do
  p:= n;
  if n < 10 then
    p1:= 0
  else
    p1:= A[floor(n/10)];
  fi;
  if isprime(p) and p > p1 then
     A[n]:= p
  else
    for d from 1 while not assigned(A[n]) do
      for i from 1 to 10^d-1 do
        p:= 10^d*n+i;
        if p > p1 and isprime(p) then
           A[n]:= p;
           break
        fi;
      od
    od
  fi
od:
seq(A[i],i=1..N); # Robert Israel, Jun 29 2015

sw(m,n)=d1=digits(n);d2=digits(m);for(i=1,min(#d1,#d2),if(d1[i]!=d2[i],return(0)));1
v=[];n=1;while(n<70,k=1;while(k<10^3,p=prime(k);if(sw(p,n)&&!vecsearch(vecsort(v),p),v=concat(v,p);k=0;n++);k++));v \\ Derek Orr, Jun 13 2015

For n>=2, a(n) >= prime(n-1). The equality holds iff prime(n-1) did not already appear as a(k), k

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A338715 Smallest prime ending with decimal expansion of n, for n relatively prime to 10.

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