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

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, 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, 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

Offset: 1

Author

Ursula Gagelmann (gagelmann(AT)altavista.net)

Keywords

• nonn
• easy
• nice

Comments

By Carmichael's conjecture, a(n) <> 1 for any n. See A074987. - Thomas Ordowski, Sep 13 2017

a(n) = 0 iff n is a term of A079695. - Bernard Schott, Oct 02 2021

Examples

			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.
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Matteo Caorsi and Sergio Cecotti, Geometric classification of 4d N=2 SCFTs, arXiv:1801.04542 [hep-th], 2018.
• Carl Pomerance, Popular values of Euler's function, Mathematica 27 (1980), 84-89.

Crossrefs

Bisection of A014197.

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

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

Programs

Magma

[#EulerPhiInverse( 2*n):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

Maple

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

Mathematica

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

PARI

a(n) = invphiNum(2*n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Extensions

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