A258190 - OEIS

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

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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

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