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aQ[n_] := PrimeQ[n] && EvenQ[Count[IntegerDigits[n, 2], 1]] == OddQ[Mod[n, 3]] && Module[{d = Reverse[IntegerDigits[n, 2]]}, Total@d[[1;; -1;; 2]] >= Total@d[[2;; -1;; 2]]]; Select[Range[5300], aQ] (* Amiram Eldar, Dec 15 2018 *)

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Primes in A065049 which are not in A139370.

277, 337, 349, 373, 853, 1093, 1109, 1117, 1237, 1297, 1301, 1303, 1361, 1367, 1373, 1381, 1399, 1429, 1489, 1493, 1621, 1861, 1873, 1877, 1879, 2389, 3413, 3541, 4177, 4357, 4373, 4421, 4423, 4441, 4447, 4549, 4561, 4567, 4597, 4933, 4951, 4957, 5077, 5189, 5197, 5209, 5233, 5237

In the notation of A139370, a prime p is in the sequence iff e(p)>o(p) and e(p)-o(p)== 4 or 5 (mod 6). [From __Vladimir Shevelev_, Dec 12 2008]

(PARI) isokp(p) = (p>2) && isprime(p) && ((hammingweight(p) % 2) != ((p % 3) % 2)); \\ A065049

isok0(n) = {my(irb = Vec(select(x->(x%2), Vecrev(binary(n)), 1))); #select(x->(x%2), irb) < #irb/2; } \\ A139370

isok(p) = isokp(p) && !isok0(p); \\ Michel Marcus, Dec 15 2018

Cf. A065049 , A139370.

Missing 853 and more terms from Michel Marcus, Dec 15 2018

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In the notation of A139370, a prime p is in the sequence iff e(p)>o(p) and e(p)-o(p)== 4 or 5 (mod 6). [From _Vladimir Shevelev (shevelev(AT)bgu.ac.il), _, Dec 12 2008]

_Vladimir Shevelev (shevelev(AT)bgu.ac.il), _, Dec 11 2008, Dec 12 2008

Primes in A065049 which are not in A139370

277, 337, 349, 373, 1093, 1109, 1117, 1237, 1297, 1301, 1303, 1361, 1367, 1373, 1381, 1399, 1429, 1489, 1493, 1621, 1861, 1873, 1877, 1879

1,1

In the notation of A139370, a prime p is in the sequence iff e(p)>o(p) and e(p)-o(p)== 4 or 5 (mod 6). [From Vladimir Shevelev (shevelev(AT)bgu.ac.il), Dec 12 2008]

Cf. A065049 A139370

nonn,new

Vladimir Shevelev (shevelev(AT)bgu.ac.il), Dec 11 2008, Dec 12 2008

approved