

A082897


Perfect totient numbers.


8



3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, 1594323, 4190263, 4782969, 9056583, 14348907, 43046721
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

It is trivial that perfect totient numbers must be odd. It is easy to show that powers of 3 are perfect totient numbers.  Ianucci
The product of the first n Fermat primes (A019434) is also a perfect totient number. There are 57 terms under 10^11.  Jud McCranie, Feb 24 2012
Terms 15, 255, 65535 and 4294967295 also belong to A051179 (see Theorem 4 in Loomis link).  Michel Marcus, Mar 19 2014


REFERENCES

L. Perez Cacho, "Sobre la suma de indicadores de ordenes sucesivos", Revista Matematica HispanoAmericana, 5.3 (1939), 4550.
A. L. Mohan and D. Suryanarayana, "Perfect totient numbers", in: Number Theory (Proc. Third Matscience Conf., Mysore, 1981) Lecture Notes in Math. 938 (SpringerVerlag, New York, 1982) pp. 101105.


LINKS

Robert G. Wilson v and Jud McCranie, Table of n, a(n) for n = 1..57 (Robert G. Wilson v produced the first 51 terms)
Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, J. Integer Sequences, 6 (2003), #03.4.5.
Paul Loomis, Michael Plytage and John Polhill, Summing up the Euler phi function, The College Mathematics Journal, Vol. 39, No. 1, Jan. 2008.
Igor E. Shparlinski, On the sum of iterations of the Euler function, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.6.


FORMULA

n is a perfect totient number if S(n) = n, where S(n) = phi(n) + phi^2(n) + ... + 1, where phi is Euler's totient function and phi^2(n) = phi(phi(n)), ..., phi^k(n) = phi(phi^(k1)(n)).
n such that n = A092693(n).
n such that 2n = A053478(n).  Vladeta Jovovic, Jul 02 2004
n log log log log n << a(n) <= 3^n.  Charles R Greathouse IV, Mar 22 2012


EXAMPLE

327 is a perfect totient number because 327 = 216 + 72 + 24 + 8 + 4 + 2 + 1. Note that 216 = phi(327), 72 = phi(216), 24 = phi(72) and so on.


MAPLE

with(numtheory):
A082897_list := proc(N) local k, p, n, L;
L := NULL;
for n from 3 by 2 to N do
k := 0; p := phi(n);
while 1 < p do k := k + p; p := phi(p) od;
if k + 1 = n then L := L, n fi
od; L end: # Peter Luschny, Nov 01 2010


MATHEMATICA

kMax = 57395631; a = Table[0, {kMax}]; PTNs = {}; Do[e = EulerPhi[k]; a[[k]] = e + a[[e]]; If[k == a[[k]], AppendTo[PTNs, k]], {k, 2, kMax}]; PTNs (* Ianucci *)
perfTotQ[n_] := Plus @@ FixedPointList[ EulerPhi@ # &, n] == 2n + 1; Select[Range[1000], perfTotQ] (* Robert G. Wilson v, Nov 06 2010 *)


PROG

(PARI) S(n)=if(n==1, 1, n=eulerphi(n); n+S(n))
for(n=2, 1e3, if(S(n)==n, print1(n", "))) \\ Charles R Greathouse IV, Mar 29 2012


CROSSREFS

Cf. A092693 (sum of iterated phi(n)). See also A091847.
Cf. A051179, A125734.
Sequence in context: A087031 A089632 A247643 * A147516 A233819 A131822
Adjacent sequences: A082894 A082895 A082896 * A082898 A082899 A082900


KEYWORD

nonn


AUTHOR

Douglas E. Iannucci (diannuc(AT)uvi.edu), Jul 21 2003


EXTENSIONS

Corrected by T. D. Noe, Mar 11 2004


STATUS

approved



