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A082897
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Perfect totient numbers.
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27
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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
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OFFSET
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1,1
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COMMENTS
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It is trivial that perfect totient numbers must be odd. It is easy to show that powers of 3 are perfect totient numbers.
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
For the first 64 terms, a(n) is approximately 1.56^n. - Jud McCranie, Jun 17 2017
These numbers were first studied in 1939 by the Spanish mathematician Laureano Pérez-Cacho Villaverde (1900-1957). The term "perfect totient number" was coined by Venkataraman (1975). - Amiram Eldar, Mar 10 2021
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B41, pp. 147-150.
L. Pérez-Cacho, Sobre la suma de indicadores de órdenes sucesivos (in Spanish), Revista Matematica Hispano-Americana, Vol.5, No. 3 (1939), pp. 45-50.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 3, pp. 240-242.
D. L. Silverman, Problem 1040, J. Recr. Math., Vol. 14 (1982); Solution by R. I. Hess, ibid., Vol. 15 (1983).
M. V. Subbarao, On a Function connected with phi(n), The Mathematics Student, Vol. 23 (1955), pp. 178-179.
T. Venkataraman, Perfect totient number, The Mathematics Student, Vol. 43 (1975), p. 178.
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LINKS
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Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, Journal of Integer Sequences, Vol. 6 (2003), Article 03.4.5.
A. L. Mohan and D. Suryanarayana, Perfect totient numbers, in: K. Alladi (ed.), Number Theory, Proceedings of the Third Matscience Conference Held at Mysore, India, June 3-6, 1981, Lecture Notes in Mathematics, Vol 938, Springer, Berlin, Heidelber, 1982, pp. 101-105.
Hans Sieburg and Michael Kentgens, On Phi-perfect numbers, in: J. Akiyama et al. (eds.), Number Theory and Combinatorics, Japan 1984, World Scientific, 1985, pp. 245-254.
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FORMULA
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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)).
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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
perfTotQ[n_] := Plus @@ FixedPointList[ EulerPhi@ # &, n] == 2n + 1; Select[Range[1000], perfTotQ] (* Robert G. Wilson v, Nov 06 2010 *)
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PROG
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(PARI) S(n)={n=eulerphi(n); if(n==1, 1, n+S(n))}
(Perl) use ntheory "euler_phi"; sub S { my $n=euler_phi(shift); return 1 if $n == 1; $n+S($n); } for (2..1e4) { say if $_==S($_); } # Dana Jacobsen, Dec 16 2018
(Python)
from itertools import count, islice
from gmpy2 import digits
from sympy import totient
def A082897_gen(startvalue=3): # generator of terms >= startvalue
for n in count((k:=max(startvalue, 3))+1-(k&1), 2):
t = digits(n, 3)
if t.count('0') == len(t)-1:
yield n
else:
m, s = n, 1
while (m:=totient(m))>1:
s += m
if s == n:
yield n
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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