A233388 -id:A233388 - cp's OEIS frontend

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

Juri-Stepan Gerasimov, Oct 06 2013

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)

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

Subsequence of A002145.
Cf. A000069, A001969, A007814, A027697, A092246, A095282.
Seems to consist of all primes in A233388.

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

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

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

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

A234011 The sums of 2 consecutive odious numbers (A000069).

Original entry on oeis.org

3, 6, 11, 15, 19, 24, 27, 30, 35, 40, 43, 47, 51, 54, 59, 63, 67, 72, 75, 79, 83, 86, 91, 96, 99, 102, 107, 111, 115, 120, 123, 126, 131, 136, 139, 143, 147, 150, 155, 160, 163, 166, 171, 175, 179, 184, 187, 191, 195, 198, 203, 207, 211, 216, 219, 222, 227, 232, 235, 239, 243, 246

Offset: 1

Gerasimov Sergey, Dec 27 2013

The union of A131323(k) and (A225822(m)+(-1)^m).
All even numbers in this sequence are evil numbers (A001969).
It seems that A233388(n) = a(A091785(n)).

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000069, A003159 (indices of odd numbers in A234011), A036554 (indices of even numbers in A234011), A131323 (odd sums of 2 successive odious or 2 successive evil numbers), A233388 (odious numbers in A234011), A234431 (sums of 2 consecutive evil numbers), A017101, A091785, A225822, A227930, A233388.

Plus @@@ Partition[Select[Range[125], OddQ[DigitCount[#, 2][[1]]] &], 2, 1] (* Amiram Eldar, Jul 24 2023 *)

a(n)=4*n-hammingweight(n-1)%2-hammingweight(n)%2 \\ Charles R Greathouse IV, Dec 29 2013

(define (A234011 n) (+ (A000069 n) (A000069 (+ n 1)))) ;; Antti Karttunen, Dec 29 2013

a(n) = A000069(n) + A000069(n + 1).
4n - 2 <= a(n) <= 4n. - Charles R Greathouse IV, Dec 29 2013
a(2n+1) = 8n + 3 = A017101(n). - Ralf Stephan, Dec 31 2013

Terms recomputed and checked by Antti Karttunen, Dec 29 2013

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

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