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%I #44 Mar 10 2021 03:18:43
%S 1,2,6,8,12,20,32,40,42,48,54,84,108,110,120,128,156,160,192,220,240,
%T 252,272,312,336,342,432,486,500,504,506,512,544,640,660,684,768,812,
%U 840,880,930,936,960,972,1000,1012,1080,1248,1320,1332,1344,1624,1632
%N 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.
%C Also numbers n such that n = y*phi(y) for a unique positive integer y (see A194507). - _Franz Vrabec_, Aug 27 2011
%C Sequence A002618 sorted into ascending order; also A327171 sorted into ascending order, with duplicate terms removed. Indices of nonzero terms in A327170 and in A327172. - _Antti Karttunen_, Sep 29 2019
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 224.
%H Amiram Eldar, <a href="/A082473/b082473.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)
%H Walther Janous, <a href="http://www.jstor.org/stable/2322406">Problem 6588</a>, Advanced Problems, The American Mathematical Monthly, Vol. 95, No. 10 (1988), p. 963; <a href="http://www.jstor.org/stable/2323872">How Often is n*phi(n) <= x^2?, Solution to Problem 6588</a>, ibid., Vol. 98, No. 5 (1991), pp. 446-448.
%F From _Antti Karttunen_, Sep 29 2019: (Start)
%F a(n) = A002618(A194507(n)).
%F A327172(a(n)) = A194507(n).
%F (End)
%F 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
%t 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 *)
%o (PARI) isok(n) = {for (x=1, n, if (eulerphi(x)*core(x) == n, return (1));); return (0);} \\ _Michel Marcus_, Dec 04 2013
%Y Cf. A000010, A002618, A007913, A194507, A327170, A327171, A327172.
%K nonn
%O 1,2
%A _Benoit Cloitre_, Apr 27 2003
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