This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
prime(1) = 2, in binary 10, has one neighbor 11 in P(2, 2), so a(1) = 1. prime(14) = 43, in binary 101011, has neighbors 101001 (41), 101111 (47), 111011 (59), so a(14) = 3.
a:= n-> (p-> nops(select(isprime, {seq(Bits[Xor]
(p, 2^i), i=0..ilog2(p)-1)})))(ithprime(n)):
seq(a(n), n=1..100); # Alois P. Heinz, May 11 2022
A352942[n_] := Count[BitXor[#, 2^Range[0, BitLength[#] - 2]], _?PrimeQ] & [Prime[n]]; Array[A352942, 100] (* Paolo Xausa, Apr 23 2025 *)
from sympy import isprime, sieve
def neighs(s):
digs = "01"
ham1 = (s[:i]+d+s[i+1:] for i in range(len(s)) for d in digs if d!=s[i])
yield from (h for h in ham1 if h[0] != '0')
def a(n):
return sum(1 for s in neighs(bin(sieve[n])[2:]) if isprime(int(s, 2)))
print([a(n) for n in range(1, 88)])
a(3) = 1 since 3 = 11_2 can be changed to 10_2 = 2, which is prime. a(5) = 1 since 5 = 101_2 can be changed to 001_2 = 1, 111_2 = 7 (prime), or 100_2 = 4. a(6) = 2 since 6 = 110_2 can be changed to 010_2 = 2 (prime), 100_2 = 4, or 111_2 = 7 (prime). a(7) = 2 since 7 = 111_2 can be changed to 011_2 = 3 (prime), 101_2 = 5 (prime), or 110_2 = 6.
a:= n-> nops(select(isprime, [seq(Bits[Xor](2^i, n), i=0..ilog2(n))])): seq(a(n), n=0..100); # Alois P. Heinz, Apr 23 2025
A383271[n_] := Count[BitXor[n, 2^Range[0, BitLength[n] - 1]], _?PrimeQ]; Array[A383271, 100, 0] (* Paolo Xausa, Apr 23 2025 *)
from gmpy2 import is_prime
def a(n):
if n == 0:
return 0
if n&1 == 0:
return int(is_prime(n + 1)) + int(1<<(n.bit_length()-1)^n == 2)
mask, c = 1, 0
for i in range(n.bit_length()):
if is_prime(mask^n):
c += 1
mask <<= 1
return c
print([a(n) for n in range(90)])
There are no 1-digit primes in base 2, so a(1) = 0. The 2-digit optimal prime ladder 10 - 11 is tied for the longest amongst 2-digit primes in binary, so a(2) = 2. The 3-digit optimal prime ladder 101 - 111 is tied for the longest amongst 3-digit primes in binary, so a(3) = 2. The only 4-digit primes in binary, 1011 and 1101, are disconnected, so a(3) = 1. The 5-digit optimal prime ladder 10001 - 10011 - 10111 - 11111 - 11101 is tied for the longest amongst 5-digit primes in binary, so a(5) = 5.
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