A381606 a(n) is the smallest prime number greater than n that contains n as a substring of its digits.

101, 11, 23, 13, 41, 53, 61, 17, 83, 19, 101, 113, 127, 113, 149, 151, 163, 173, 181, 191, 1201, 211, 223, 223, 241, 251, 263, 127, 281, 229, 307, 131, 1321, 233, 347, 353, 367, 137, 383, 139, 401, 241, 421, 431, 443, 457, 461, 347, 487, 149, 503, 151, 521, 353, 541

Offset: 0

Author

Joost de Winter, Mar 01 2025

Keywords

• nonn
• base

Examples

			The first prime number greater than 0 that contains "0" is 101, so a(0) = 101.
The first prime number greater than 1 that contains "1" is 11, so a(1) = 11.
The first prime number greater than 2 that contains "2" is 23, so a(2) = 23.
		

Links

• Joost de Winter, Table of n, a(n) for n = 0..9000
• David A. Corneth, PARI program
• Joost de Winter, MATLAB function

Crossrefs

Cf. A062584, A354114, A381099, A064735.

Programs

MATLAB

\\ See De Winter link

Maple

f:= proc(n) local m,d,d1,x,y,L;
  m:= length(n);
  for d from 1 do
    L:= sort([seq(10^d * n + x, x = 1 .. 10^d-1, 2),
             seq(n+10^m*x, x=10^(d-1) .. 10^d-1),
             seq(seq(seq(10^d1*n + x + 10^(m+d1)*y, x=1 .. 10^d1-1,2),y=10^(d-d1-1) .. 10^(d-d1)-1),d1=1..d-1)]);
    for x in L do if isprime(x) then return x fi od
  od
end proc:
f(0):= 101:
map(f, [$0..100]); # Robert Israel, Mar 02 2025

Mathematica

a[n_] := Module[{p = NextPrime[n + 1], s = ToString[n]}, While[! StringContainsQ[ToString[p], s], p = NextPrime[p]]; p]; Array[a, 100, 0] (* Amiram Eldar, Mar 03 2025 *)

PARI

a(n) = my(p=nextprime(n+1), s=Str(n)); while (#strsplit(Str(p), s) < 2, p = nextprime(p+1)); p; \\ Michel Marcus, Mar 01 2025

PARI

\\ See Corneth link

Formula

a(n) > 2n. For large enough n, a(n) < n^5 by the strongest known version of Linnik's theorem. - Charles R Greathouse IV, Mar 01 2025