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A082473
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Numbers n such that n = phi(x)*core(x) for some x <= n, where phi(x) is the Euler totient function and core(x) the squarefree part of x.
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7
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1, 2, 6, 8, 12, 20, 32, 40, 42, 48, 54, 84, 108, 110, 120, 128, 156, 160, 192, 220, 240, 252, 272, 312, 336, 342, 432, 486, 500, 504, 506, 512, 544, 640, 660, 684, 768, 812, 840, 880, 930, 936, 960, 972, 1000, 1012, 1080, 1248, 1320, 1332, 1344, 1624, 1632
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OFFSET
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1,2
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COMMENTS
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Also numbers n such that n = y*phi(y) for a unique positive integer y (see A194507). - Franz Vrabec, Aug 27 2011
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REFERENCES
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József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 224.
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LINKS
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Walther Janous, Problem 6588, Advanced Problems, The American Mathematical Monthly, Vol. 95, No. 10 (1988), p. 963; How Often is n*phi(n) <= x^2?, Solution to Problem 6588, ibid., Vol. 98, No. 5 (1991), pp. 446-448.
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FORMULA
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(End)
The number of terms not exceeding x is ~ c * sqrt(x), where c = Product_{p prime} (1 + 1/sqrt(p*(p-1)) - 1/p) = 1.3651304521... (Janous, 1988). - Amiram Eldar, Mar 10 2021
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MATHEMATICA
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With[{nn = 1700}, TakeWhile[Union@ Array[EulerPhi[#] (Sqrt@ # /. (c_: 1) a_^(b_: 0) :> (c a^b)^2) &, nn], # <= nn &]] (* Michael De Vlieger, Sep 29 2019, after Bill Gosper at A007913 *)
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PROG
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(PARI) isok(n) = {for (x=1, n, if (eulerphi(x)*core(x) == n, return (1)); ); return (0); } \\ Michel Marcus, Dec 04 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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