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 A237840 a(n) = |{0 < k <= n: the number of twin prime pairs not exceeding k*n is a square}|. 5
 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, 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, 2, 4, 4, 1, 3, 2, 4, 2, 4, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 for no n > 159. (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. We have verified that a(n) > 0 for all n = 1, ..., 22000. 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. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..9000 Zhi-Wei Sun, A surprising conjecture on primes and squares, a message to the Number Theory List, Feb. 14, 2014. Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014. EXAMPLE a(4) = 1 since the number of twin prime pairs not exceeding 1*4 = 4 is 0^2. 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}). a(18055) > 0 since there are exactly 675^2 = 455625 twin prime pairs not exceeding 5758*18055 = 103960690. a(18120) > 0 since there are exactly 729^2 = 531441 twin prime pairs not exceeding 6827*18120 = 123705240. a(18307) > 0 since there are exactly 681^2 = 463761 twin prime pairs not exceeding 5792*18307 = 106034144. a(18670) > 0 since there are exactly 683^2 = 466489 twin prime pairs not exceeding 5716*18670 = 106717720. a(19022) > 0 since there are exactly 737^2 = 543169 twin prime pairs not exceeding 6666*19022 = 126800652. a(19030) > 0 since there are exactly 706^2 = 498436 twin prime pairs not exceeding 6045*19030 = 115036350. a(19805) > 0 since there are exactly 717^2 = 514089 twin prime pairs not exceeding 6015*19805 = 119127075. a(19939) > 0 since there are exactly 1000^2 = 10^6 twin prime pairs not exceeding 12660*19939 = 252427740. a(20852) > 0 since there are exactly 747^2 = 558009 twin prime pairs not exceeding 6268*20852 = 130700336. a(21642) > 0 since there are exactly 724^2 = 524176 twin prime pairs not exceeding 5628*21642 = 121801176. MATHEMATICA tw[0]:=0 tw[n_]:=tw[n-1]+If[PrimeQ[Prime[n]+2], 1, 0] SQ[n_]:=IntegerQ[Sqrt[tw[PrimePi[n]]]] a[n_]:=Sum[If[SQ[k*n-2], 1, 0], {k, 1, n}] Table[a[n], {n, 1, 80}] CROSSREFS Cf. A000290, A001359, A005384, A006512, A237706, A237710, A237578, A237598, A237612, A237769, A237817, A237839, A237879, A237975. Sequence in context: A097289 A294207 A288267 * A114115 A126268 A193739 Adjacent sequences: A237837 A237838 A237839 * A237841 A237842 A237843 KEYWORD nonn AUTHOR Zhi-Wei Sun, Feb 14 2014 STATUS approved

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Last modified August 12 19:26 EDT 2024. Contains 375113 sequences. (Running on oeis4.)