cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A296655 Numbers k such that phi(x) = k has a positive even number of solutions.

This page as a plain text file.
%I A296655 #34 Jul 22 2021 02:13:06
%S A296655 1,4,6,10,12,16,18,22,24,28,30,36,42,46,52,54,58,64,66,70,78,80,82,84,
%T A296655 88,100,102,106,110,112,126,130,136,138,148,150,160,162,166,168,172,
%U A296655 176,178,180,184,190,196,198,200,208,210,222,224,226,228,232,238,250
%N A296655 Numbers k such that phi(x) = k has a positive even number of solutions.
%C A296655 When the number of solutions is 2, the sum of Sum_{d|x} d*mu(d) is always 0.
%C A296655 A007366 is contained in this sequence because it selects terms with the smallest even number of inverses.
%F A296655 0 = card({phi^-1(a(n))}) mod 2.
%e A296655 1 is a term because phi(1) has two inverses, 1, and 2.
%e A296655 6 is a term because phi(6) has four inverses, 7, 9, 14, and 18.
%e A296655 10 is a term because phi(10) has two inverses, 11, and 22.
%e A296655 18 is a term because phi(18) has four inverses, 19, 27, 38, 54.
%e A296655 348 is a term because phi(348) has six inverses, 349, 413, 531, 698, 826, and 1062.
%t A296655 With[{nn = 1500}, TakeWhile[Union@ Select[KeyValueMap[{#1, Length@ #2} &, PositionIndex@ Array[EulerPhi, nn]], EvenQ@ Last@ # &][[All, 1]], # <= nn/6 &] ] (* _Michael De Vlieger_, Dec 20 2017 *)
%Y A296655 Cf. A007366, A014197, A032446, A071386.
%K A296655 nonn
%O A296655 1,2
%A A296655 _Torlach Rush_, Dec 17 2017
%E A296655 Corrected and extended by _Rémy Sigrist_, Dec 19 2017