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A346064 Number of primes that may be generated by changing any two digits of n simultaneously.

%I A346064 #22 Aug 17 2021 19:24:45
%S A346064 21,17,20,15,21,20,21,16,21,17,23,18,22,17,23,22,23,17,23,19,23,19,22,
%T A346064 16,23,22,23,18,23,18,22,18,21,16,22,21,22,17,22,17,23,18,22,17,23,22,
%U A346064 23,17,23,19,23,19,22,16,23,22,23,18,23,18,22,18,21,16,22
%N A346064 Number of primes that may be generated by changing any two digits of n simultaneously.
%C A346064 Indices of low records are given by A345289. By heuristic considerations it is conjectured that a(n) > 0 for all n >= 10.
%H A346064 M. Filaseta, M. Kozek, Ch. Nicol and J. Selfridge, <a href="https://people.math.sc.edu/filaseta/papers/FKNSpaper0808.pdf">Composites that Remain Composite After Changing a Digit</a>, J. Comb. Number Theory, Vol. 2, No. 1 (2010), pp. 25-36.
%e A346064 Changing two digits of the number 17 simultaneously yields the primes 02,03,05,23,29,31,41,43,53,59,61,71,73,79,83,89, so a(17) = 16.
%p A346064 A346064 := proc(n)
%p A346064 local a, d, e, r, s, l, N, NN, nn, i;
%p A346064 a := 0;
%p A346064 N := convert(n, base, 10);
%p A346064 l := nops(N);
%p A346064 for d to l - 1 do
%p A346064    for e from d + 1 to l do
%p A346064       for r from 0 to 9 do
%p A346064          for s from 0 to 9 do
%p A346064         if r <> op(d, N) and s <> op(e, N) then
%p A346064            NN := subsop(d = r, e = s, N);
%p A346064            nn := add(op(i, NN)*10^(i - 1), i = 1 .. l);
%p A346064            if isprime(nn) then a := a + 1; end if;
%p A346064         end if;
%p A346064         end do;
%p A346064       end do;
%p A346064    end do;
%p A346064 end do;
%p A346064 a;
%p A346064 end proc:
%t A346064 Table[Count[Flatten[FromDigits/@Tuples[ReplacePart[t=List/@IntegerDigits[n],{#->Complement[Range[0,9],t[[#]]],#2->Complement[Range[0,9],t[[#2]]]}]&@@#]&/@Subsets[Range@IntegerLength@n,{2}]],_?PrimeQ],{n,10,100}] (* _Giorgos Kalogeropoulos_, Jul 23 2021 *)
%o A346064 (Python)
%o A346064 from sympy import isprime
%o A346064 from itertools import combinations, product
%o A346064 def change2(s):
%o A346064     for i, j in combinations(range(len(s)), 2):
%o A346064         for c, d in product("0123456789", repeat=2):
%o A346064             if c != s[i] and d != s[j]:
%o A346064                 yield s[:i] + c + s[i+1:j] + d + s[j+1:]
%o A346064 def a(n): return sum(isprime(int(t)) for t in change2(str(n)))
%o A346064 print([a(n) for n in range(10, 101)]) # _Michael S. Branicky_, Jul 23 2021
%Y A346064 Cf. A345289 (indices of record lows).
%Y A346064 Cf. A209252 (changing one digit).
%K A346064 nonn,base
%O A346064 10,1
%A A346064 _Franz Vrabec_, Jul 03 2021