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A140746
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Numbers n such that n^2 + 3 is powerful, (i.e., is of the form a^2*b^3, with a>=1, b>=1).
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1
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OFFSET
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1,2
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COMMENTS
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Florian Luca proved that this sequence is infinite, by showing that 37*x(7*k) + 98*y(7*k) is in the sequence, where x(k) = A001081(k) and y(k) = A001080(k) are solutions of the Pell equation x^2 - 7*y^2 = 1. The sequence of these numbers is 37, 9667939010, 2524807950507510523, 659360302164952911361460078, ... - Amiram Eldar, Aug 22 2018
a(7) <= 457189690981. - Giovanni Resta, Aug 23 2018
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REFERENCES
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J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 37, pp 14, Ellipses, Paris 2008.
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LINKS
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Table of n, a(n) for n=1..6.
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EXAMPLE
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37 is the sequence since 37^2 + 3 = 1372 = 2^2 * 7^3 is powerful.
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MATHEMATICA
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powerfulQ[n_] := Min@FactorInteger[n][[All, 2]] > 1; Select[Range[100000], powerfulQ[#^2 + 3] &] (* Amiram Eldar, Aug 22 2018 *)
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PROG
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(PARI) isok(n) = vecmin(factor(n^2+3)[, 2]) > 1; \\ Michel Marcus, Aug 24 2018
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CROSSREFS
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Cf. A001694 (powerful), A001080, A001081, A117950 (n^2+3).
Sequence in context: A015061 A015037 A262786 * A350307 A262646 A063681
Adjacent sequences: A140743 A140744 A140745 * A140747 A140748 A140749
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KEYWORD
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nonn,more
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AUTHOR
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Lekraj Beedassy, Jul 12 2008
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EXTENSIONS
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a(5) corrected and a(6) removed by Amiram Eldar, Aug 22 2018
a(6) from Giovanni Resta, Aug 23 2018
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STATUS
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approved
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