A091847 -id:A091847 - cp's OEIS frontend

A082897 Perfect totient numbers.

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

Offset: 1

Douglas E. Iannucci, Jul 21 2003

nonn

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

			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.

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).

Jud McCranie, Table of n, a(n) for n = 1..64 (first 51 terms from Robert G. Wilson v)
Jovele G. Belmonte, On perfect totient numbers, Masteral Thesis, De La Salle University, 2006.
Li-Xia Dai and Yong-Gao Chen, A note on perfect totient numbers, Journal of Northeast Normal University, Vol. 39, No. 4 (2007), pp. 17-19.
Tuukka Hyvärinen, Täydelliset totienttiluvut (in Finnish), Master's thesis, Tampere University, 2015; alternative link.
Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On Perfect Totient Numbers, Journal of Integer Sequences, Vol. 6 (2003), Article 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.
Florian Luca, On the Distribution of Perfect Totients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.4.
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.
Deng Moujie, A Note On Perfect Totient Numbers, Journal of Integer Sequences, Vol. 12 (2009), Article 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.
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.
M. V. Subbarao, On a Function connected with phi(n), The Mathematics Student, Vol. 23 (1955), pp. 178-179; entire volume.
T. Venkataraman, Perfect totient number, The Mathematics Student, Vol. 43, No. 2 (1975), p. 178; entire issue.
Wikipedia, Perfect totient number.

Cf. A092693 (sum of iterated phi(n)). See also A091847.

Cf. A051179, A125734.

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

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 *)

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

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

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
A082897_list = list(islice(A082897_gen(),20)) # Chai Wah Wu, Mar 24 2023

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

Corrected by T. D. Noe, Mar 11 2004

A092693 Sum of iterated phi(n).

Original entry on oeis.org

0, 1, 3, 3, 7, 3, 9, 7, 9, 7, 17, 7, 19, 9, 15, 15, 31, 9, 27, 15, 19, 17, 39, 15, 35, 19, 27, 19, 47, 15, 45, 31, 35, 31, 39, 19, 55, 27, 39, 31, 71, 19, 61, 35, 39, 39, 85, 31, 61, 35, 63, 39, 91, 27, 71, 39, 55, 47, 105, 31, 91, 45, 55, 63, 79, 35, 101, 63, 79, 39, 109, 39, 111

Offset: 1

T. D. Noe, Mar 04 2004

nonn

Iannucci, Moujie and Cohen examine perfect totient numbers: n such that a(n) = n.

			a(100) = 71 because the iterations of phi (40, 16, 8, 4, 2, 1) sum to 71.

T. D. Noe, Table of n, a(n) for n = 1..10000
C. Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1
P. Erdos and M. V. Subbarao, On the iterates of some arithmetic functions, The theory of arithmetic functions (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich. 1971), Lecture Notes in Math., 251 , pp. 119-125, Springer, Berlin, 1972. [alternate link]
Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, On perfect totient numbers, J. Integer Sequences, 6 (2003), #03.4.5.

Cf. A003434 (iterations of phi(n) needed to reach 1), A092694 (iterated phi product).

Cf. A082897 and A091847 (perfect totient numbers).

a092693 1 = 0
a092693 n = (+ 1) $ sum $ takeWhile (/= 1) $ iterate a000010 $ a000010 n
-- Reinhard Zumkeller, Oct 27 2011

nMax=100; a=Table[0, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e+a[[e]], {n, 2, nMax}]; a (* T. D. Noe *)
Table[Plus @@ FixedPointList[EulerPhi, n] - (n + 1), {n, 72}] (* Alonso del Arte, Jan 29 2007 *)

a(n)=my(k);while(n>1,k+=n=eulerphi(n));k \\ Charles R Greathouse IV, Mar 22 2012

from sympy import totient
from math import prod
def f(n):
    m = n
    while m > 1:
        m = totient(m)
        yield m
def A092693(n): return sum(f(n)) # Chai Wah Wu, Nov 14 2021

a(1) = 0, a(n) = phi(n) + a(phi(n))
a(n) = A053478(n) - n. - Vladeta Jovovic, Jul 02 2004
Erdős & Subbarao prove that a(n) ~ phi(n) for almost all n. In particular, a(n) < n for almost all n. The proportion of numbers up to N for which a(n) > n is at most 1/log log log log N. - Charles R Greathouse IV, Mar 22 2012

A286233 Amicable totient numbers: pairs of numbers (m, n) such that n = A092693(m) and m = A092693(n).

Original entry on oeis.org

579, 639, 14911, 18207, 38575, 47223, 310399, 492855, 16632919, 20238207, 34696495, 37400607, 37852351, 52463103, 84250111, 122992023

Offset: 1

Amiram Eldar, May 04 2017

Analogous to amicable pairs (A063990) as perfect totient numbers (A082897) are analogous to perfect numbers (A000396).
The sequence lists the numbers in increasing order. The first 8 pairs (m, n) are adjacent to each other in the list.
No other terms below 10^9.

			A092693(579) = phi(579) + phi(phi(579)) + ... = 384 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 639, and A092693(639) = phi(639) + phi(phi(639)) + ... = 420 + 96 + 32 + 16 + 8 + 4 + 2 + 1 = 579.

Cf. A092693, A082897, A091847.

totSum[n_] := Plus @@ FixedPointList[EulerPhi@# &, n] - n - 1; amicableTotQ[n_] := If[Nest[totSum, n, 2] == n && totSum[n] != n, True, False]; Select[Range[10^9], amicableTotQ[#] &]

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A082897 Perfect totient numbers.

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A092693 Sum of iterated phi(n).

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A286233 Amicable totient numbers: pairs of numbers (m, n) such that n = A092693(m) and m = A092693(n).

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