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

21, 17, 20, 15, 21, 20, 21, 16, 21, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22, 21, 22, 17, 22, 17, 23, 18, 22, 17, 23, 22, 23, 17, 23, 19, 23, 19, 22, 16, 23, 22, 23, 18, 23, 18, 22, 18, 21, 16, 22

Offset: 10

Author

Franz Vrabec, Jul 03 2021

Keywords

• nonn
• base

Comments

Indices of low records are given by A345289. By heuristic considerations it is conjectured that a(n) > 0 for all n >= 10.

Examples

			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.
		

Links

• M. Filaseta, M. Kozek, Ch. Nicol and J. Selfridge, Composites that Remain Composite After Changing a Digit, J. Comb. Number Theory, Vol. 2, No. 1 (2010), pp. 25-36.

Crossrefs

Cf. A345289 (indices of record lows).

Cf. A209252 (changing one digit).

Programs

Maple

A346064 := proc(n)
local a, d, e, r, s, l, N, NN, nn, i;
a := 0;
N := convert(n, base, 10);
l := nops(N);
for d to l - 1 do
   for e from d + 1 to l do
      for r from 0 to 9 do
         for s from 0 to 9 do
        if r <> op(d, N) and s <> op(e, N) then
           NN := subsop(d = r, e = s, N);
           nn := add(op(i, NN)*10^(i - 1), i = 1 .. l);
           if isprime(nn) then a := a + 1; end if;
        end if;
        end do;
      end do;
   end do;
end do;
a;
end proc:

Mathematica

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 *)

Python

from sympy import isprime
from itertools import combinations, product
def change2(s):
    for i, j in combinations(range(len(s)), 2):
        for c, d in product("0123456789", repeat=2):
            if c != s[i] and d != s[j]:
                yield s[:i] + c + s[i+1:j] + d + s[j+1:]
def a(n): return sum(isprime(int(t)) for t in change2(str(n)))
print([a(n) for n in range(10, 101)]) # Michael S. Branicky, Jul 23 2021