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

0,1

Links

Joost de Winter, Table of n, a(n) for n = 0..9000

David A. Corneth, PARI program

Joost de Winter, MATLAB function

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

Example

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.

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 *)

Prog

(MATLAB) \\ See De Winter link
(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

Crossrefs

Cf. A062584, A354114, A381099, A064735.

Sequence in context: A369956 A381099 A385512 * A029948 A174926 A029949

Adjacent sequences: A381603 A381604 A381605 * A381607 A381608 A381609

Keyword

nonn,base

Author

Joost de Winter, Mar 01 2025

Status

approved