A209252 Number of primes (excluding n) that may be generated by replacing any decimal digit of n with a digit from 0 to 9.

4, 4, 3, 3, 4, 3, 4, 3, 4, 4, 4, 7, 5, 9, 4, 5, 4, 8, 4, 7, 2, 7, 3, 7, 2, 3, 2, 8, 2, 5, 2, 5, 3, 9, 2, 3, 2, 6, 2, 7, 3, 6, 4, 8, 3, 4, 3, 7, 3, 8, 2, 7, 3, 7, 2, 3, 2, 8, 2, 5, 2, 5, 3, 9, 2, 3, 2, 6, 2, 7, 3, 6, 4, 8, 3, 4, 3, 9, 3, 6, 2, 7, 3, 7, 2, 3, 2, 8, 2, 5, 1, 6, 2, 8, 1, 2, 1, 5, 1, 6, 4, 10, 5, 9, 4

Offset: 0

Author

Michel Lagneau, Jan 14 2013

Keywords

• nonn
• base

Comments

I expect that the average value of a(n) is 45/log 100 if n is coprime to 10 and 0 otherwise. - Charles R Greathouse IV, Jan 14 2013

First occurrence of k = 0..27: 200, 90, 20, 2, 1, 12, 37, 11, 17, 13, 101, 109, 107, 177, 357, 1001, 1011, 10759, 13299, 11487, 42189, 113183, 984417, 344253, 1851759, 4787769, 16121457, 15848679. - Robert G. Wilson v, Dec 19 2015

The number of prime neighbors of n in H(A055642(n), 10), where H(k,b) is the Hamming graph whose vertices are the sequences of length k over the alphabet {0,1,...,b-1} with adjacency being defined by having Hamming distance 1 (see A158576). - Michael S. Branicky, Apr 22 2025

Examples

			a(0) = 4 because by replacing the digit 0, we obtain the 4 primes 2, 3, 5 and 7;
a(11) = 7 because by replacing the 1st digit of *1, we obtain the primes 31, 41, 61, 71, and by replacing the 2nd digit of 1* we obtain the primes 13, 17, 19, hence a(11) = 7.
a(13) = 8 because 03, 11, 17, 19, 23, 43, 53, 73 and 83 are all primes.
a(204) = 0 because it is impossible to find a prime number if we replace the digits 2, 0 or 4.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 0..10000
• Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.

Crossrefs

Cf. A000040, A055642, A158576.

Programs

Maple

A209252 := proc(n)
    local a,dgs,d,r,pd,p ;
    a := 0 ;
    dgs := convert(n,base,10) ;
    for d from 1 to nops(dgs) do
        for r from 0 to 9 do
            pd := subsop(d=r,dgs) ;
            p := add(op(i,pd)*10^(i-1),i=1..nops(pd)) ;
            if isprime(p) and p <> n then
                a := a+1 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Jan 18 2013

Mathematica

f[n_] := Block[{c = k = 0, d, p, lmt = 1 + Floor@ Log10@ n}, While[k < lmt, d = 0; While[d < 10, p = Quotient[n, 10^(k+1)]*10^(k+1) + d*10^k + Mod[n, 10^k]; If[p != n && PrimeQ@ p, c++]; d++]; k++]; c]; f[0] = 4; Array[f, 105, 0] (* Robert G. Wilson v, Dec 19 2015 *)

Python

from sympy import isprime
def A209252(n):
    return len([1 for i in range(len(str(n))) for d in '0123456789' if d != str(n)[i] and isprime(int(str(n)[:i]+d+str(n)[i+1:]))]) # Chai Wah Wu, Sep 19 2016

Python

from gmpy2 import digits, is_prime
def a(n):
    s, c = list(map(int, digits(n))), 0
    if len(s) > 1 and s[-1] not in {1, 3, 7, 9}:
        z = int(is_prime(s[-1])) if all(c == 0 for c in s[1:-1]) else 0
        return z + sum(1 for e in {1, 3, 7, 9} if is_prime(n + e - s[-1]))
    for i in range(len(s)):
        b = 10**(len(s)-1-i)
        for j in range(10):
            if j != s[i]:
                t = n + (j-s[i])*b
                if is_prime(t):
                    c += 1
    return c
print([a(n) for n in range(100)]) # Michael S. Branicky, Apr 22 2025