A352942 Let p = prime(n); a(n) = number of primes q with same number of binary digits as p that can be obtained from p by changing one binary digit.

1, 1, 1, 1, 0, 0, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 3, 2, 0, 1, 1, 0, 2, 1, 2, 3, 1, 1, 4, 1, 0, 1, 1, 0, 1, 3, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 2, 2, 0, 3, 2, 1, 1, 2, 2, 1, 1, 0, 3, 0, 0, 2, 2, 0, 2, 2, 2, 3, 2, 2, 0, 2, 0, 1, 2, 0, 1

Offset

1,8

Comments

a(n) is also the degree of prime(n) in the graph P(A070939(prime(n)), 2), defined in A145667.

Links

Table of n, a(n) for n=1..87.

Formula

a(n) = deg(prime(n)) in P(A070939(prime(n)), 2) (see A145667).

Example

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.

Maple

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

Prog

(Python)
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)])

Crossrefs

Binary analog of A125002.

Cf. A000040, A004676, A070939, A104080, A014234, A137985, A145667, A353738.

Sequence in context: A087775 A366091 A089955 * A180312 A178819 A369174

Adjacent sequences: A352939 A352940 A352941 * A352943 A352944 A352945

Keyword

nonn,base

Author

Michael S. Branicky, May 11 2022

Status

approved