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A358154 a(n) is the smallest composite number obtained by appending one or more 1's to n.

%I A358154 #21 Dec 22 2022 14:40:21
%S A358154 111,21,3111,411,51,611,711,81,91,1011,111,121,1311,141,15111,161,171,
%T A358154 18111,1911,201,21111,221,231,24111,2511,261,27111,2811,291,301,3111,
%U A358154 321,3311,341,351,361,371,381,391,4011,411,42111,4311,441,451,4611,471,481,4911,501,511,5211,531,5411
%N A358154 a(n) is the smallest composite number obtained by appending one or more 1's to n.
%C A358154 a(n) is either 10*n+1, 100*n+11 or 1000*n+111, because at least one of these is divisible by 3. - _Robert Israel_, Nov 01 2022
%C A358154 Actually: exactly one of these is divisible by 3. Almost all terms are a(n) = 10n + 1: this is the case for about (k-1)/k of the terms up to 10^k (i.e., 69% ~ 2/3 up to 10^3, 76% = 3/4 up to 10^4, 80% = 4/5 up to 10^5, 83% = 5/6 up to 10^6). - _M. F. Hasler_, Nov 03 2022
%p A358154 f:= proc(n) local x;
%p A358154    x:= n;
%p A358154    do
%p A358154      x:= 10*x+1;
%p A358154      if not isprime(x) then return x fi;
%p A358154    od
%p A358154 end proc:
%p A358154 map(f, [$1..100]); # _Robert Israel_, Nov 01 2022
%t A358154 a[n_] := NestWhile[10*# + 1 &, 10*n + 1, ! CompositeQ[#] &]; Array[a, 54] (* _Amiram Eldar_, Nov 01 2022 *)
%o A358154 (Python)
%o A358154 from sympy import isprime
%o A358154 def A358154(n):
%o A358154     t = str(n)+'1'
%o A358154     while isprime(int(t)):t=t+'1'
%o A358154     return int(t)
%o A358154 print([A358154(i) for i in range(1, 100)])
%o A358154 (PARI) a(n) = my(d=digits(n), m); if (!isprime(n), d = concat(d, 1)); while(isprime(m=fromdigits(d)), d=concat(d, 1)); m; \\ _Michel Marcus_, Nov 01 2022
%Y A358154 Cf. A069568, A112386, A002808, A153275.
%K A358154 nonn,base
%O A358154 1,1
%A A358154 _Gleb Ivanov_, Nov 01 2022