A237840 - OEIS

%I #46 Mar 16 2014 11:59:48

%S 1,2,2,1,3,3,2,1,1,2,1,2,2,1,2,1,1,2,1,2,3,1,3,4,3,4,4,3,3,4,2,2,4,2,

%T 3,2,1,3,2,2,2,1,2,1,2,2,2,4,3,2,2,1,3,4,3,1,3,1,2,4,2,5,2,3,2,3,1,3,

%U 2,4,4,1,3,2,4,2,4,4,4,4

%N a(n) = |{0 < k <= n: the number of twin prime pairs not exceeding k*n is a square}|.

%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 159.

%C (ii) For every n = 1, 2, 3, ..., there is a positive integer k <= n such that the number |{{p, 2*p+1}: both p and 2*p + 1 are primes not exceeding k*n}| is a square.

%C We have verified that a(n) > 0 for all n = 1, ..., 22000.

%C See also A237879 for the least k among 1, ..., n such that the number of twin prime pairs not exceeding k*n is a square.

%H Zhi-Wei Sun, <a href="/A237840/b237840.txt">Table of n, a(n) for n = 1..9000</a>

%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;68a81953.1402">A surprising conjecture on primes and squares</a>, a message to the Number Theory List, Feb. 14, 2014.

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

%e a(4) = 1 since the number of twin prime pairs not exceeding 1*4 = 4 is 0^2.

%e a(9) = 1 since there are exactly 2^2 twin prime pairs not exceeding 3*9 = 27 (namely, they are {3, 5}, {5, 7}, {11, 13} and {17, 19}).

%e a(18055) > 0 since there are exactly 675^2 = 455625 twin prime pairs not exceeding 5758*18055 = 103960690.

%e a(18120) > 0 since there are exactly 729^2 = 531441 twin prime pairs not exceeding 6827*18120 = 123705240.

%e a(18307) > 0 since there are exactly 681^2 = 463761 twin prime pairs not exceeding 5792*18307 = 106034144.

%e a(18670) > 0 since there are exactly 683^2 = 466489 twin prime pairs not exceeding 5716*18670 = 106717720.

%e a(19022) > 0 since there are exactly 737^2 = 543169 twin prime pairs not exceeding 6666*19022 = 126800652.

%e a(19030) > 0 since there are exactly 706^2 = 498436 twin prime pairs not exceeding 6045*19030 = 115036350.

%e a(19805) > 0 since there are exactly 717^2 = 514089 twin prime pairs not exceeding 6015*19805 = 119127075.

%e a(19939) > 0 since there are exactly 1000^2 = 10^6 twin prime pairs not exceeding 12660*19939 = 252427740.

%e a(20852) > 0 since there are exactly 747^2 = 558009 twin prime pairs not exceeding 6268*20852 = 130700336.

%e a(21642) > 0 since there are exactly 724^2 = 524176 twin prime pairs not exceeding 5628*21642 = 121801176.

%t tw[0]:=0

%t tw[n_]:=tw[n-1]+If[PrimeQ[Prime[n]+2],1,0]

%t SQ[n_]:=IntegerQ[Sqrt[tw[PrimePi[n]]]]

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

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

%Y Cf. A000290, A001359, A005384, A006512, A237706, A237710, A237578, A237598, A237612, A237769, A237817, A237839, A237879, A237975.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Feb 14 2014