cp's OEIS Frontend

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

%I A209252 #45 Apr 23 2025 02:38:19
%S A209252 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,
%T A209252 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,
%U A209252 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
%N A209252 Number of primes (excluding n) that may be generated by replacing any decimal digit of n with a digit from 0 to 9.
%C A209252 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
%C A209252 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
%C A209252 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
%H A209252 Michel Lagneau, <a href="/A209252/b209252.txt">Table of n, a(n) for n = 0..10000</a>
%H A209252 Terence Tao, <a href="http://arxiv.org/abs/0802.3361">A remark on primality testing and decimal expansions</a>, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
%e A209252 a(0) = 4 because by replacing the digit 0, we obtain the 4 primes 2, 3, 5 and 7;
%e A209252 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.
%e A209252 a(13) = 8 because 03, 11, 17, 19, 23, 43, 53, 73 and 83 are all primes.
%e A209252 a(204) = 0 because it is impossible to find a prime number if we replace the digits 2, 0 or 4.
%p A209252 A209252 := proc(n)
%p A209252     local a,dgs,d,r,pd,p ;
%p A209252     a := 0 ;
%p A209252     dgs := convert(n,base,10) ;
%p A209252     for d from 1 to nops(dgs) do
%p A209252         for r from 0 to 9 do
%p A209252             pd := subsop(d=r,dgs) ;
%p A209252             p := add(op(i,pd)*10^(i-1),i=1..nops(pd)) ;
%p A209252             if isprime(p) and p <> n then
%p A209252                 a := a+1 ;
%p A209252             end if;
%p A209252         end do:
%p A209252     end do:
%p A209252     a ;
%p A209252 end proc: # _R. J. Mathar_, Jan 18 2013
%t A209252 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 *)
%o A209252 (Python)
%o A209252 from sympy import isprime
%o A209252 def A209252(n):
%o A209252     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
%o A209252 (Python)
%o A209252 from gmpy2 import digits, is_prime
%o A209252 def a(n):
%o A209252     s, c = list(map(int, digits(n))), 0
%o A209252     if len(s) > 1 and s[-1] not in {1, 3, 7, 9}:
%o A209252         z = int(is_prime(s[-1])) if all(c == 0 for c in s[1:-1]) else 0
%o A209252         return z + sum(1 for e in {1, 3, 7, 9} if is_prime(n + e - s[-1]))
%o A209252     for i in range(len(s)):
%o A209252         b = 10**(len(s)-1-i)
%o A209252         for j in range(10):
%o A209252             if j != s[i]:
%o A209252                 t = n + (j-s[i])*b
%o A209252                 if is_prime(t):
%o A209252                     c += 1
%o A209252     return c
%o A209252 print([a(n) for n in range(100)]) # _Michael S. Branicky_, Apr 22 2025
%Y A209252 Cf. A000040, A055642, A158576.
%K A209252 nonn,base
%O A209252 0,1
%A A209252 _Michel Lagneau_, Jan 14 2013