A308575 - OEIS

%I #21 Jun 24 2019 14:25:12

%S 2,2946,4,18,830,86,342,498,36002,2310,14660,3791908,138060,160110,

%T 998836,4345842,357341648,56717562,36609556,5972021576,2654687244,

%U 8237027666,22719286202,1542163060562,222365303318

%N a(n) is the least positive even number k such that among the first k prime numbers there are exactly k/2 prime numbers where the n-th least significant bit is one, or a(n) = -1 if no such k exists.

%C Is a(n) always positive?

%C If a(n) > 0, then a(n) >= 2*A000720(2^(n-1)-1). - _Chai Wah Wu_, Jun 13 2019

%F When a(n) > 0, Sum_{k = 1..a(n)} (-1)^floor(prime(k)/2^(n-1)) = 0 (where prime(k) denotes the k-th prime number).

%o (PARI) { s = vector(18); a = vector(#s); u = 1; forprime (p=2, oo, n++; for (b=1, #s, if (!a[b], s[b]+=(-1)^bittest(p,b-1); if (s[b]==0, a[b]=n; while (a[u], print1 (a[u]", "); u++; if (u>#a, break(3))))))) }

%o (Python)

%o from sympy import primepi

%o def A308575(n):

%o     n2, t1 = 2**(n-1), 0

%o     k = n2 - 1

%o     kp = primepi(k)

%o     kp2 = primepi(k+n2)-kp

%o     while kp2 < kp or t1 >= kp:

%o         k += n2

%o         t1, t2 = kp, kp2

%o         kp2 = primepi(k+n2) - kp2

%o         kp = t2

%o     return 2*kp # _Chai Wah Wu_, Jun 13 2019

%Y Cf. A000040, A000720.

%K nonn,base,more,hard

%O 1,1

%A _Rémy Sigrist_, Jun 08 2019

%E a(20)-a(23) from _Chai Wah Wu_, Jun 13 2019

%E a(24)-a(25) from _Chai Wah Wu_, Jun 24 2019