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

1,1

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

Formula

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

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

Prog

(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

Crossrefs

Cf. A000040, A045572, A258083, A338715.

Sequence in context: A334132 A245193 A338715 * A244471 A083968 A082769

Adjacent sequences: A258187 A258188 A258189 * A258191 A258192 A258193

Keyword

nonn,base

Author

Vladimir Shevelev, May 23 2015

Status

approved