A238881 - OEIS

%I #17 Sep 21 2015 04:05:30

%S 0,1,1,0,0,2,1,3,1,2,2,5,2,3,2,4,3,5,2,3,2,4,1,8,4,4,4,3,2,5,2,4,4,4,

%T 4,8,2,4,2,5,4,6,3,2,6,6,3,11,6,10,4,8,2,11,4,7,4,7,2,12,4,6,2,6,3,8,

%U 3,5,8,12,6,12,4,15,8,11,5,12,2,11

%N Number of odd primes p < 2*n with  prime(n*(p+1)/2) + n*(p+1)/2 prime.

%C Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 2, 3, 7, 9, 23. Moreover, for any r = 1,-1 and n > 5*(2+r) there is a positive integer k < n such that 2*k+r and prime(k*n)+k*n are both prime.

%C (ii) If n > 1 is not equal to 13, then prime(k*n) - k*n is prime for some k = 1, ..., n.

%C This conjecture implies that there are infinitely many positive integers m with prime(m) + m (or prime(m) - m) prime.

%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169--187. (See Conjecture 3.21(i) and note that the typo 2k+1 there should be 2k-1.)

%H Zhi-Wei Sun, <a href="/A238881/b238881.txt">Table of n, a(n) for n = 1..7000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e a(7) = 1 since 11 and prime(7*(11+1)/2) + 7*(11+1)/2 = prime(42) + 42 = 181 + 42 = 223 are both prime.

%e a(23) = 1 since 7 and prime(23*(7+1)/2) + 23*(7+1)/2 = prime(92) + 92 = 479 + 92 = 571 are both prime.

%t PQ[n_]:=PrimeQ[Prime[n]+n]

%t p[k_,n_]:=PQ[(Prime[k]+1)/2*n]

%t a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]

%t Table[a[n],{n,1,80}]

%o (PARI) a(n) = {my(nb = 0); forprime(p=3, 2*n, if (isprime(prime(n*(p+1)/2) + n*(p+1)/2), nb++);); nb;} \\ _Michel Marcus_, Sep 21 2015

%Y Cf. A000040, A064269, A064402, A238573, A238576, A238878.

%K nonn

%O 1,6

%A _Zhi-Wei Sun_, Mar 06 2014