A227930 Primes p such that p-1 and p+1 have an even Hamming weight.

11, 19, 47, 59, 67, 79, 107, 131, 179, 191, 211, 227, 239, 251, 271, 283, 307, 331, 367, 379, 419, 431, 443, 463, 491, 499, 563, 587, 659, 719, 787, 827, 859, 883, 911, 947, 971, 1019, 1039, 1051, 1087, 1123, 1163, 1171, 1187, 1231, 1259, 1279, 1291, 1327, 1423, 1451, 1459, 1471, 1483

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 06 2013

Keywords

• nonn
• base
• easy

Comments

Primes such that both neighbors are evil (as defined in A001969).

From Antti Karttunen, Dec 29 2013: (Start)

Excluding 2, the intersection of A027697 (Odious primes: primes with odd number of 1's in binary expansion) and A095282 (Primes whose binary-expansion ends with an even number of 1's).

Equally, the intersection of A092246 (Odd "odious" numbers) and A095282.

Equally, odd odious primes p such that A007814(p+1) is even.

(End)

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A002145.

Cf. A000069, A001969, A007814, A027697, A092246, A095282.

Seems to consist of all primes in A233388.

Programs

Maple

read("transforms"):
isA000069 := proc(n)
    if wt(n) mod 2 = 1 then
        true;
    else
        false;
    end if;
end proc:
for n from 1 do
    if isprime(n) and not isA000069(n-1) and not isA000069(n+1) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 08 2013
(Scheme, with Antti Karttunen's IntSeq-library, two alternative implementations)
(define A227930 (MATCHING-POS 1 1 (lambda (n) (and (even? (A000120 (- n 1))) (even? (A000120 (+ n 1))) (prime? n)))))
(define A227930v2 (MATCHING-POS 1 1 (lambda (n) (and (odd? n) (odd? (A000120 n)) (even? (A007814 (+ n 1))) (prime? n))))) # Antti Karttunen, Dec 29 2013

Mathematica

Select[Prime[Range[250]], And @@ EvenQ[DigitCount[# + {-1, 1}, 2, 1]] &] (* Amiram Eldar, Jul 24 2023 *)

PARI

is(n)=hammingweight(n-1)%2==0 && hammingweight(n+1)%2==0 && isprime(n) \\ Charles R Greathouse IV, Oct 09 2013

Extensions

Entries checked by R. J. Mathar, Oct 08 2013