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A323745 a(n) is the smallest prime that becomes composite if any single digit of its base-n expansion is changed to a different digit (but not to zero).

%I A323745 #22 Mar 25 2024 16:37:15
%S A323745 3,2,89,67,28151,223,6211,2789,294001,701,8399011,2423,691063,243367,
%T A323745 527099,10513,2078920243,10909,169402249,2114429,156760543,68543,
%U A323745 96733308587,181141,121660507,6139219,3141223681,114493
%N A323745 a(n) is the smallest prime that becomes composite if any single digit of its base-n expansion is changed to a different digit (but not to zero).
%C A323745 This sequence has several terms in common with A186995; if the restriction that no digit can be changed to zero were removed, A186995 would result.
%C A323745 a(30) > 10^10.
%C A323745 The next few terms after a(30) are 356479, 860343287, 399946711, ...
%F A323745 a(n) <= A186995(n). - _Chai Wah Wu_, Mar 25 2024
%e A323745 a(2)=3 because 3 is prime and its base-2 expansion is 11_2, which cannot have either of its digits changed to a nonzero digit, whereas the only smaller prime, i.e., 2 = 10_2, yields another prime if its 0 digit is changed to a 1.
%e A323745 a(3)=2 because 2 = 2_3 is prime and, in base 3, the only way to change its digit to another (nonzero) digit is to change it to 1_3 = 1, which is nonprime.
%e A323745 a(4)=89 because 89 = 1121_4 is prime, every number that can be obtained by changing one of its digits to another (nonzero) digit is nonprime (1122_4=90, 1123_4=91, 1111_4=85, 1131_4=93, 1221_4=105, 1321_4=121, 2121_4=153, 3121_4=217), and there is no prime smaller than 89 that has this property.
%e A323745 a(18)=2078920243 because 2078920243 = 3723de91_18 (where the letters d and e represent the base-18 digits whose values are 13 and 14, respectively), and each of the 128 other base-18 numbers that can be obtained by changing one of its eight digits to another (nonzero) digit is nonprime, and no smaller prime has this property.
%o A323745 (Python)
%o A323745 from sympy import isprime, nextprime
%o A323745 from sympy.ntheory import digits
%o A323745 def A323745(n):
%o A323745     p = 2
%o A323745     while True:
%o A323745         m = 1
%o A323745         for j in digits(p,n)[:0:-1]:
%o A323745             for k in range(1,n):
%o A323745                 if k!=j and isprime(p+(k-j)*m):
%o A323745                     break
%o A323745             else:
%o A323745                 m *= n
%o A323745                 continue
%o A323745             break
%o A323745         else:
%o A323745             return p
%o A323745         p = nextprime(p) # _Chai Wah Wu_, Mar 25 2024
%Y A323745 Cf. A186995.
%K A323745 nonn,base,more
%O A323745 2,1
%A A323745 _Jon E. Schoenfield_, May 04 2019