A082473 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.

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

Offset: 1

Author

Benoit Cloitre, Apr 27 2003

Keywords

• nonn

Comments

Also numbers n such that n = y*phi(y) for a unique positive integer y (see A194507). - Franz Vrabec, Aug 27 2011

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

References

• József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, p. 224.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
• 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.

Crossrefs

Cf. A000010, A002618, A007913, A194507, A327170, A327171, A327172.

Programs

Mathematica

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 *)

PARI

isok(n) = {for (x=1, n, if (eulerphi(x)*core(x) == n, return (1));); return (0);} \\ Michel Marcus, Dec 04 2013

Formula

From Antti Karttunen, Sep 29 2019: (Start)

a(n) = A002618(A194507(n)).

A327172(a(n)) = A194507(n).

(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