cp's OEIS Frontend

A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.

%I A383271 #27 Apr 23 2025 19:31:05
%S A383271 0,0,1,1,1,1,2,2,0,2,2,1,1,1,0,3,1,1,2,3,0,4,1,3,0,2,0,3,1,2,1,2,0,2,
%T A383271 1,2,1,2,0,3,1,1,1,4,0,5,1,1,0,2,0,2,1,2,0,2,0,3,1,1,1,2,0,4,0,3,2,3,
%U A383271 0,3,1,4,1,1,0,5,0,4,1,1,0,4,1,2,0,0,0,3,1,1
%N A383271 Number of primes (excluding n) that may be generated by replacing any binary digit of n with a digit from 0 to 1.
%C A383271 Also, the number of prime neighbors of n in H(A070939(n), 2), 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 A145667).
%C A383271 Prepending 1 is not allowed.
%H A383271 Michael S. Branicky, <a href="/A383271/b383271.txt">Table of n, a(n) for n = 0..10000</a>
%e A383271 a(3) = 1 since 3 = 11_2 can be changed to 10_2 = 2, which is prime.
%e A383271 a(5) = 1 since 5 = 101_2 can be changed to 001_2 = 1, 111_2 = 7 (prime), or 100_2 = 4.
%e A383271 a(6) = 2 since 6 = 110_2 can be changed to 010_2 = 2 (prime), 100_2 = 4, or 111_2 = 7 (prime).
%e A383271 a(7) = 2 since 7 = 111_2 can be changed to 011_2 = 3 (prime), 101_2 = 5 (prime), or 110_2 = 6.
%p A383271 a:= n-> nops(select(isprime, [seq(Bits[Xor](2^i, n), i=0..ilog2(n))])):
%p A383271 seq(a(n), n=0..100);  # _Alois P. Heinz_, Apr 23 2025
%t A383271 A383271[n_] := Count[BitXor[n, 2^Range[0, BitLength[n] - 1]], _?PrimeQ];
%t A383271 Array[A383271, 100, 0] (* _Paolo Xausa_, Apr 23 2025 *)
%o A383271 (Python)
%o A383271 from gmpy2 import is_prime
%o A383271 def a(n):
%o A383271     if n == 0:
%o A383271         return 0
%o A383271     if n&1 == 0:
%o A383271         return int(is_prime(n + 1)) + int(1<<(n.bit_length()-1)^n == 2)
%o A383271     mask, c = 1, 0
%o A383271     for i in range(n.bit_length()):
%o A383271         if is_prime(mask^n):
%o A383271             c += 1
%o A383271         mask <<= 1
%o A383271     return c
%o A383271 print([a(n) for n in range(90)])
%Y A383271 Base-2 analog of A209252.
%Y A383271 Cf. A070939, A145667, A352942.
%K A383271 nonn,base
%O A383271 0,7
%A A383271 _Michael S. Branicky_, Apr 21 2025