A258190 Smallest prime not appearing earlier that ends with A045572(n).

11, 3, 7, 19, 211, 13, 17, 419, 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, 4127, 1129, 131, 4133, 137, 4139, 2141, 2143, 5147, 11149, 1151, 1153, 4157, 4159, 2161, 1163, 167, 3169

Offset: 1

Author

Vladimir Shevelev, May 23 2015

Keywords

• nonn
• base

Comments

Using Dirichlet's theorem, we conclude that every term exists. So the sequence is a permutation of the odd primes other than 5. Indeed, an odd prime p other than 5 is either in its natural place in A045572 or appears earlier than that.

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..1000
• Index entries for primes involving decimal expansion of n

Crossrefs

Cf. A000040, A045572, A258083, A338715.

Programs

Maple

r:= -1: Used:= 'Used':
for n from 1 to 1000 do
  r:= r+2;
  if r mod 5 = 0 then r:= r+2 fi;
  d:= 10^(1+ilog10(r));
  for x from r by d do
    if isprime(x) and not assigned(Used[x]) then
      a[n]:= x;
      Used[x]:= true;
      break
    fi
  od
od:
seq(a[n],n=1..1000); # Robert Israel, May 27 2015

PARI

\\with first line from A045572 by Charles R Greathouse IV
a(n) = {n = 10*(n>>2)+[-1, 1, 3, 7][n%4+1]; my(d = digits(n),m = matrix(#d + 1, 2), w=0); m[1,2] = d[#d] - 10; for(i = 2, matsize(m)[1], m[i,1]=10^(i-2)*d[#d-i+2] + m[i-1,1]; if(m[i-1,1] == m[i,1],m[i,2]=m[i-1,2], j=m[i,1]==m[i-1,2]; while(!isprime(10^(i-1)*j+m[i,1]), j++); m[i,2]=10^(i-1)*j+m[i,1]));m[matsize(m)[1],2]} \\ David A. Corneth, May 25 2015

Python

from sympy import isprime
def aupton(terms):
    alst, aset = [], set()
    for n in range(1, terms+1):
        ending = 2*n - 1 + (n+1)//4 * 2 # A045572
        i, pow10 = ending, 10**len(str(ending))
        while i in aset or not isprime(i): i += pow10
        alst.append(i); aset.add(i)
    return alst
print(aupton(68)) # Michael S. Branicky, Nov 03 2021

Formula

a(n) >= A045572(n). The equality holds iff A045572(n) is a prime that did not already appear as a(k), k

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