A063439 a(n) = phi(n)^phi(n).

1, 1, 4, 4, 256, 4, 46656, 256, 46656, 256, 10000000000, 256, 8916100448256, 46656, 16777216, 16777216, 18446744073709551616, 46656, 39346408075296537575424, 16777216, 8916100448256, 10000000000, 341427877364219557396646723584, 16777216, 104857600000000000000000000

Offset: 1

Author

Reinhard Zumkeller, Jul 24 2001

Keywords

• nonn

Comments

Number of endofunctions over RRS[n]. Used in proof of Dirichlet theorem to derive characters: a(n) = A000312(A000010(n)). - Labos Elemer, May 27 2002

Sum_{n>=1} 1/phi(n)^phi(n) ~ 2.765711032... and so apparently equals Sum_{n>=1} A014197(n)/n^n where A014197(n) is the number of numbers m such that phi(m) = n. Is this a known result? - Gerald McGarvey, May 16 2004

The equality above is true and it is not unique for phi: for each value k = phi(n) the summand 1/phi(n)^phi(n) appears A014197(k) times, so Sum_{n>=1} 1/phi(n)^phi(n) = Sum_{k>=1} A014197(k) * (1/k^k). - Amiram Eldar, Dec 10 2024

Links

• Harry J. Smith, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000010, A000312, A014197.

Programs

Magma

[EulerPhi(n)^EulerPhi(n): n in [1..30]]; // Vincenzo Librandi, Dec 29 2019

Mathematica

Table[EulerPhi[n]^EulerPhi[n], {n, 30}] (* Vincenzo Librandi, Dec 29 2019 *)

PARI

{ for (n=1, 200, p=eulerphi(n); write("b063439.txt", n, " ", p^p) ) } \\ Harry J. Smith, Aug 21 2009

Formula

n log n / (log log n) << log a(n) < n log n. - Charles R Greathouse IV, Jan 19 2012