A032446 - OEIS

%I #60 Nov 15 2024 05:21:57

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

%C By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - _Thomas Ordowski_, Sep 13 2017

%C a(n) = 0 iff n is a term of A079695. - _Bernard Schott_, Oct 02 2021

%D Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Second Edition, Dover Publications, Inc., NY, 1966, page 90.

%H T. D. Noe, <a href="/A032446/b032446.txt">Table of n, a(n) for n = 1..5000</a>

%H Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).

%H 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 Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/popular.pdf">Popular values of Euler's function</a>, Mathematica 27 (1980), 84-89.

%e 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 with(numtheory); [ seq(nops(invphi(2*n)), n=1..90) ];

%t t = Table[0, {100} ]; Do[a = EulerPhi[n]; If[a < 202, t[[a/2]]++ ], {n, 3, 10^5} ]; t

%o (Magma) [#EulerPhiInverse( 2*n):n in [1..100]]; // _Marius A. Burtea_, Sep 08 2019

%o (PARI) a(n) = invphiNum(2*n); \\ _Amiram Eldar_, Nov 15 2024 using _Max Alekseyev_'s invphi.gp

%Y Bisection of A014197.

%Y Cf. A000010, A005277, A074987, A079695, A085758.

%Y Cf. A006511 (largest k for which A000010(k) = A002202(n)), A057635.

%K nonn,easy,nice

%O 1,1

%A Ursula Gagelmann (gagelmann(AT)altavista.net)

%E Extended by Robin Trew (trew(AT)hcs.harvard.edu).