A354114 -id:A354114 - cp's OEIS frontend

A354113 The smallest number that contains all the digits of n in order but does not equal n.

10, 10, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153

Offset: 0

Jianing Song, May 17 2022

If n begins with 1, then a(n) is obtained by inserting a 0 after it; otherwise a(n) is obtained by placing a 1 before n.

			The smallest number not equal to 19 containing the digits 1 and 9 in that order is 109, so a(19) = 109.
The smallest number not equal to 22 containing two 2's is 122, so a(22) = 122.

Jianing Song, Table of n, a(n) for n = 0..10000

Cf. A354049, A354114.

a(n) = if(n==0, 10, my(k=logint(n,10)); if(n<2*10^k, n+9*10^k, n+10^(k+1)))

def a(n):
    if n == 0: return 10
    s = str(n)
    return int(s[0]+"0"+s[1:]) if s[0] == "1" else int("1"+s)
print([a(n) for n in range(54)]) # Michael S. Branicky, May 17 2022

If 10^k <= n < 2*10^k, a(n) = n + 9*10^k; if 2*10^k <= n < 10^(k+1), a(n) = n + 10^(k+1).

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

Original entry on oeis.org

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

Joost de Winter, Mar 01 2025

			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.

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

Cf. A062584, A354114, A381099, A064735.

MATLAB
```
\\ See De Winter link
```

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

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 *)

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
```

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

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A354113 The smallest number that contains all the digits of n in order but does not equal n.

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

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