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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 Hispano-Americana, 5.3 (1939), 45-50.

A. L. Mohan and D. Suryanarayana, "Perfect totient numbers", in: Number Theory (Proc. Third Matscience Conf., Mysore, 1981) Lecture Notes in Math. 938 (Springer-Verlag, New York, 1982) pp. 101-105.

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^(k-1)(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: A055927 A087031 A089632 * 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

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Last modified September 2 02:37 EDT 2014. Contains 246321 sequences.