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

1,1

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

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.

Example

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.

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

Prog

(Magma) [#EulerPhiInverse( 2*n):n in [1..100]]; // Marius A. Burtea, Sep 08 2019
(PARI) a(n) = invphiNum(2*n); \\ Amiram Eldar, Nov 15 2024 using Max Alekseyev's invphi.gp

Crossrefs

Bisection of A014197.

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

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

Sequence in context: A185271 A352285 A158012 * A271563 A342938 A028949

Adjacent sequences: A032443 A032444 A032445 * A032447 A032448 A032449

Keyword

nonn,easy,nice,changed

Author

Ursula Gagelmann (gagelmann(AT)altavista.net)

Extensions

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

Status

approved