cp's OEIS Frontend

A032446 Number of solutions to phi(k) = 2n.

%I A032446 #60 Nov 15 2024 05:21:57
%S A032446 3,4,4,5,2,6,0,6,4,5,2,10,0,2,2,7,0,8,0,9,4,3,2,11,0,2,2,3,2,9,0,8,2,
%T A032446 0,2,17,0,0,2,10,2,6,0,6,0,3,0,17,0,4,2,3,2,9,2,6,0,3,0,17,0,0,2,9,2,
%U A032446 7,0,2,2,3,0,21,0,2,2,0,0,7,0,12,4,3,2,12,0,2,0,8,2,10
%N A032446 Number of solutions to phi(k) = 2n.
%C A032446 By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - _Thomas Ordowski_, Sep 13 2017
%C A032446 a(n) = 0 iff n is a term of A079695. - _Bernard Schott_, Oct 02 2021
%D A032446 Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover Publications, Inc., NY, 1966, page 90.
%H A032446 T. D. Noe, <a href="/A032446/b032446.txt">Table of n, a(n) for n = 1..5000</a>
%H A032446 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A032446 Matteo Caorsi and Sergio Cecotti, <a href="https://arxiv.org/abs/1801.04542">Geometric classification of 4d N=2 SCFTs</a>, arXiv:1801.04542 [hep-th], 2018.
%H A032446 Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/popular.pdf">Popular values of Euler's function</a>, Mathematica 27 (1980), 84-89.
%e A032446 If n = 8 then phi(x) = 2*8 = 16 is satisfied for only a(8) = 6 values of x, viz. 17, 32, 34, 40, 48, 60.
%p A032446 with(numtheory); [ seq(nops(invphi(2*n)), n=1..90) ];
%t A032446 t = Table[0, {100} ]; Do[a = EulerPhi[n]; If[a < 202, t[[a/2]]++ ], {n, 3, 10^5} ]; t
%o A032446 (Magma) [#EulerPhiInverse( 2*n):n in [1..100]]; // _Marius A. Burtea_, Sep 08 2019
%o A032446 (PARI) a(n) = invphiNum(2*n); \\ _Amiram Eldar_, Nov 15 2024 using _Max Alekseyev_'s invphi.gp
%Y A032446 Bisection of A014197.
%Y A032446 Cf. A000010, A005277, A074987, A079695, A085758.
%Y A032446 Cf. A006511 (largest k for which A000010(k) = A002202(n)), A057635.
%K A032446 nonn,easy,nice
%O A032446 1,1
%A A032446 Ursula Gagelmann (gagelmann(AT)altavista.net)
%E A032446 Extended by Robin Trew (trew(AT)hcs.harvard.edu).