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A334838
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Positive integers m with prime(m) in the form x^2 + m*y^2, where x and y are positive integers.
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1
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1, 2, 12, 35, 37, 77, 97, 100, 118, 136, 137, 152, 183, 184, 190, 212, 231, 258, 290, 352, 421, 462, 482, 487, 690, 730, 741, 960, 1110, 1111, 1168, 1169, 1227, 1285, 1328, 1396, 1417, 1621, 2074, 2119, 2318, 2578, 2603, 2652, 2707, 2726, 2737, 2772, 2776, 2788, 2803, 2853, 2857, 2865, 2882, 2892, 3035, 3176, 3199, 3245
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OFFSET
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1,2
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COMMENTS
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Conjecture: The current sequence has infinitely many terms.
This was first mentioned in Remark 2.21 of the linked 2017 paper.
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LINKS
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Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.)
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EXAMPLE
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a(2) = 2 with prime(2) = 3 = 1^2 + 2*1^2.
a(3) = 12 with prime(12) = 37 = 5^2 + 12*1^2.
a(4) = 35 with prime(35) = 149 = 3^2 + 35*2^2.
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MATHEMATICA
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SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={}; Do[Do[If[SQ[Prime[m]-m*x^2], tab=Append[tab, m]; Goto[aa]], {x, 1, Sqrt[Prime[m]/m]}]; Label[aa], {m, 1, 3245}]; tab
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CROSSREFS
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Cf. A000040, A000290, A232174.
Sequence in context: A055699 A244378 A062094 * A200543 A246426 A353503
Adjacent sequences: A334835 A334836 A334837 * A334839 A334840 A334841
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KEYWORD
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nonn
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AUTHOR
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Zhi-Wei Sun, May 13 2020
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STATUS
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approved
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