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A082897
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Perfect totient numbers.
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22
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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
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REFERENCES
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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.
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LINKS
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Jud McCranie, Table of n, a(n) for n = 1..64 (first 51 terms from Robert G. Wilson v)
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.
D. Moujie, A Note On Perfect Totient Numbers, JIS 12 (2009) #09.6.2.
Igor E. Shparlinski, On the sum of iterations of the Euler function, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.6.
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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)).
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
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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
od; L end: # Peter Luschny, Nov 01 2010
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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))}
for(n=2, 1e3, if(S(n)==n, print1(n", "))) \\ Charles R Greathouse IV, Mar 29 2012; Corrected by Dana Jacobsen, Dec 16 2018
(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
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CROSSREFS
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Cf. A092693 (sum of iterated phi(n)). See also A091847.
Cf. A051179, A125734.
Sequence in context: A247643 A287351 A256388 * A147516 A233819 A131822
Adjacent sequences: A082894 A082895 A082896 * A082898 A082899 A082900
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KEYWORD
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nonn
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AUTHOR
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Douglas E. Iannucci, Jul 21 2003
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EXTENSIONS
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Corrected by T. D. Noe, Mar 11 2004
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STATUS
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approved
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