A378506 The number of solutions to the equation phi(phi(x)) = n, where phi is the Euler totient function.

5, 8, 0, 13, 0, 4, 0, 23, 0, 2, 0, 14, 0, 0, 0, 36, 0, 2, 0, 5, 0, 2, 0, 32, 0, 0, 0, 2, 0, 0, 0, 54, 0, 0, 0, 11, 0, 0, 0, 23, 0, 0, 0, 5, 0, 0, 0, 66, 0, 0, 0, 2, 0, 4, 0, 3, 0, 0, 0, 2, 0, 0, 0, 78, 0, 0, 0, 0, 0, 0, 0, 38, 0, 0, 0, 0, 0, 0, 0, 41, 0, 2, 0, 6

Offset: 1

Author

Amiram Eldar, Nov 29 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• David M. Bressoud, A Course in Computational Number Theory (web page), CNT.m, Computational Number Theory Mathematica package.

Crossrefs

Cf. A000010, A010554, A014197, A378507, A378508.

Programs

Mathematica

a[n_] := Sum[PhiMultiplicity[i], {i, PhiInverse[n]}]; Array[a, 100] (* using David M. Bressoud's CNT.m *)

PARI

a(n) = vecsum(apply(x -> invphiNum(x), invphi(n))); \\ using Max Alekseyev's invphi.gp

Formula

a(n) > 0 if and only if n is in A378508.