A092693 -id:A092693 - cp's OEIS frontend

A343243 Sociable totient numbers of order 3: numbers k such that s(s(s(k))) = k, but s(k) != k, where s(k) = A092693(k) is the sum of iterated phi function.

20339, 21159, 23883, 35503, 43255, 45375, 365599, 476343, 493047, 746383, 979839, 1097367, 3331135, 3816831, 3972543, 57720703, 68705247, 78376959, 3031407415, 3742563231, 3866214695

Offset: 1

Amiram Eldar, Apr 08 2021

The numbers k such that s(k) = k are the perfect totient numbers (A082897).

a(22) > 2*10^10, if it exists.

			20339 is a term since s(20339) = 23883, s(23883) = 21159 and s(21159) = 20339.

Cf. A000010, A082897, A092693, A286233.

totSum[n_] := Plus @@ FixedPointList[EulerPhi, n] - n - 1; soc3TotQ[n_] := Nest[totSum, n, 3] == n && totSum[n] != n; Select[Range[2, 10^6], soc3TotQ]

A082897 Perfect totient numbers.

Original entry on oeis.org

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

A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

Original entry on oeis.org

1, 3, 6, 7, 12, 9, 16, 15, 18, 17, 28, 19, 32, 23, 30, 31, 48, 27, 46, 35, 40, 39, 62, 39, 60, 45, 54, 47, 76, 45, 76, 63, 68, 65, 74, 55, 92, 65, 78, 71, 112, 61, 104, 79, 84, 85, 132, 79, 110, 85, 114, 91, 144, 81, 126, 95, 112, 105, 164, 91, 152, 107, 118, 127, 144, 101

Offset: 1

Labos Elemer, Jan 14 2000

nonn

For n = 2^w, the sum is 2^(w+1) - 1.

			If phi is applied repeatedly to n = 91, the iterates {91, 72, 24, 8, 4, 2, 1} are obtained. Their sum is a(91) = 91 + 72 + 24 + 8 + 4 + 2 + 1 = 202.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000010, A010554, A049099, A049100, A049107, A049108, A032358.

Row sums of A375478.

a053478 = (+ 1) . sum . takeWhile (/= 1) . iterate a000010
-- Reinhard Zumkeller, Oct 27 2011

f[n_] := Plus @@ Drop[ FixedPointList[ EulerPhi, n], -1]; Table[ f[n], {n, 66}] (* Robert G. Wilson v, Dec 16 2004 *)
f[1] := 1; f[n_] := n + f[EulerPhi[n]]; Table[f[n], {n, 66}] (* Carlos Eduardo Olivieri, May 26 2015 *)

a(n)=my(s=n);while(n>1,s+=n=eulerphi(n)); s \\ Charles R Greathouse IV, Feb 21 2013

a(n) = n + a(phi(n)).

a(n) = A092693(n) + n. - Vladeta Jovovic, Jul 02 2004

A331273 Sum of the iterated exponential totient function (A072911).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Feb 25 2020

nonn

Analogous to A092693 with the exponential totient function ephi instead of the Euler totient function phi (A000010).

a(n) = 1 for n > 1 which is cubefree (A004709) and a(n) > 1 for n in A046099.

			a(8) = ephi(8) + ephi(ephi(8)) = 2 + 1 = 3 (where ephi is A072911).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004709, A046099, A072911, A092693.

ephi[n_] := Times @@ EulerPhi[FactorInteger[n][[;; , 2]]]; s[n_] := Plus @@ FixedPointList[ephi, n] - n - 1; Array[s, 100]

A092694 Product of iterated phi(n).

Original entry on oeis.org

1, 1, 2, 2, 8, 2, 12, 8, 12, 8, 80, 8, 96, 12, 64, 64, 1024, 12, 216, 64, 96, 80, 1760, 64, 1280, 96, 216, 96, 2688, 64, 1920, 1024, 1280, 1024, 1536, 96, 3456, 216, 1536, 1024, 40960, 96, 4032, 1280, 1536, 1760, 80960, 1024, 4032, 1280, 32768, 1536, 79872, 216

Offset: 1

T. D. Noe, Mar 04 2004

A logarithmic plot of this sequence shows an unusual banded structure.

			a(100) = 40960 because the iterations of phi (40, 16, 8, 4, 2, 1) have a product of 40960.

T. D. Noe, Table of n, a(n) for n=1..10000
T. D. Noe, Plot of A092694

Cf. A003434 (iterations of phi(n) needed to reach 1), A092693 (iterated phi sum).

Cf. A000010.

a092694 n = snd $ until ((== 1) . fst) f (a000010 n, 1) where
   f (x, p) = (a000010 x, p * x)
-- Reinhard Zumkeller, Jan 30 2014

nMax=100; a=Table[1, {nMax}]; Do[e=EulerPhi[n]; a[[n]]=e*a[[e]], {n, 2, nMax}]; a

from sympy import totient
from math import prod
def f(n):
    m = n
    while m > 1:
        m = totient(m)
        yield m
def A092694(n): return prod(f(n)) # Chai Wah Wu, Nov 14 2021

a(1) = 1, a(n) = phi(n) * a(phi(n))

A288452 Pseudoperfect totient numbers: numbers n such that equal the sum of a subset of their iterated phi(n).

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 97, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 131, 137, 139, 141, 143, 149, 151, 153, 155

Offset: 1

Amiram Eldar, Jun 09 2017

nonn

Analogous to A005835 (pseudoperfect numbers) as A082897 (perfect totient numbers) is analogous to A000396 (perfect numbers).
All the odd primes are in this sequence.
Number of terms < 10^k: 4, 40, 350, 2956, 24842, etc. - Robert G. Wilson v, Jun 17 2017
All terms are odd. If n is even, phi(n) <= n/2, and except for n = 2, we will have phi(n) also even. So the sum of the phi sequence < n*(1/2 + 1/4 + ...) = n. - Franklin T. Adams-Watters, Jun 25 2017

			The iterated phi of 25 are 20, 8, 4, 2, 1 and 25 = 20 + 4 + 1.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Supersequence of A082897. Subsequence of A286265.

Cf. A000010, A000396, A005835, A053478, A092693.

pseudoPerfectTotQ[n_]:= Module[{tots = Most[Rest[FixedPointList[EulerPhi@# &, n]]]}, MemberQ[Total /@ Subsets[tots, Length[tots]], n]]; Select[Range[155], pseudoPerfectTotQ]

subsetSum(v, target)=if(setsearch(v,target), return(1)); if(#v<2, return(target==0)); my(u=v[1..#v-1]); if(target>v[#v] && subsetSum(u, target-v[#v]), return(1)); subsetSum(u,target);
is(n)=if(isprime(n), return(n>2)); my(v=List(),k=n); while(k>1, listput(v,k=eulerphi(k))); subsetSum(Set(v),n) \\ Charles R Greathouse IV, Jun 25 2017

A329153 Sum of the iterated unitary totient function (A047994).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 16, 24, 10, 20, 9, 21, 9, 24, 39, 55, 24, 42, 21, 21, 20, 42, 23, 47, 21, 47, 42, 70, 24, 54, 85, 41, 55, 47, 47, 83, 42, 47, 70, 110, 21, 63, 54, 117, 42, 88, 54, 102, 47, 117, 83, 135, 47, 110, 63, 83, 70, 128, 47, 107, 54, 102, 165, 102

Offset: 1

Amiram Eldar, Feb 25 2020

nonn

Analogous to A092693 with the unitary totient function uphi instead of the Euler totient function phi (A000010).

			a(4) = uphi(4) + uphi(uphi(4)) + uphi(uphi(uphi(4))) = 3 + 2 + 1 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003271, A047994, A049865, A092693, A225172, A225173, A286067.

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); Table[Plus @@ FixedPointList[uphi, n] - n - 1, {n, 1, 100}]

a(n) = n for n in A286067.

A385745 The sum of the iterated infinitary analog of the totient function A384247 when started at n.

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 10, 18, 10, 20, 9, 21, 9, 18, 33, 49, 18, 36, 21, 21, 20, 42, 18, 42, 21, 36, 36, 64, 18, 48, 49, 41, 49, 42, 42, 78, 36, 42, 49, 89, 21, 63, 48, 81, 42, 88, 48, 96, 42, 81, 78, 130, 36, 89, 42, 78, 64, 122, 42, 102, 48, 96, 96, 96, 41, 107

Offset: 1

Amiram Eldar, Jul 08 2025

			  n | iterations            | a(n)
  --+-----------------------+--------------------
  2 | 2 -> 1                | 1
  3 | 3 -> 2 -> 1           | 2 + 1 = 3
  4 | 4 -> 3 -> 2 -> 1      | 3 + 2 + 1 = 6
  5 | 5 -> 4 -> 3 -> 2 -> 1 | 4 + 3 + 2 + 1 = 10
  6 | 6 -> 2 -> 1           | 2 + 1 = 3

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A384247, A385744, A385746, A385747.

Similar sequences: A092693, A329153, A333611.

f[p_, e_] := p^e*(1 - 1/p^(2^(IntegerExponent[e, 2]))); iphi[1] = 1; iphi[n_] := iphi[n] = Times @@ f @@@ FactorInteger[n];
a[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

iphi(n) = {my(f = factor(n)); n * prod(i = 1, #f~, (1 - 1/f[i, 1]^(1 << valuation(f[i, 2], 2)))); }
a(n) = if(n ==  1, 0, my(i = iphi(n)); i + a(i));

A333611 Sum of the iterated infinitary totient function iphi (A091732).

Original entry on oeis.org

0, 1, 3, 6, 10, 3, 9, 6, 14, 10, 20, 9, 21, 9, 14, 29, 45, 14, 32, 21, 21, 20, 42, 9, 33, 21, 45, 32, 60, 14, 44, 29, 41, 45, 33, 33, 69, 32, 33, 21, 61, 21, 63, 44, 61, 42, 88, 44, 92, 33, 61, 69, 121, 45, 61, 32, 69, 60, 118, 33, 93, 44, 92, 106, 92, 41, 107

Offset: 1

Amiram Eldar, Mar 28 2020

nonn

			a(3) = iphi(3) + iphi(iphi(3)) = 2 + 1 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A091732, A092693, A329153, A330273, A331273, A333609.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) - 1); a[n_] := Plus @@ NestWhileList[iphi, n, # != 1 &] - n; Array[a, 100]

A333871 Sum of the iterated absolute Möbius divisor function (A173557).

Original entry on oeis.org

0, 1, 3, 1, 5, 3, 9, 1, 3, 5, 15, 3, 15, 9, 9, 1, 17, 3, 21, 5, 15, 15, 37, 3, 5, 15, 3, 9, 37, 9, 39, 1, 25, 17, 27, 3, 39, 21, 27, 5, 45, 15, 57, 15, 9, 37, 83, 3, 9, 5, 33, 15, 67, 3, 45, 9, 39, 37, 95, 9, 69, 39, 15, 1, 51, 25, 91, 17, 59, 27, 97, 3, 75, 39

Offset: 1

Amiram Eldar, Apr 08 2020

nonn

Analogous to A092693 with the absolute Möbius divisor function (A173557) instead of the Euler totient function phi (A000010).

			a(3) = A173557(3) + A173557(A173557(3)) = 2 + 1 = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Iterating the Sum of Möbius Divisor Function and Euler Totient Function, Mathematics, Vol. 7, No. 11 (2019), pp. 1083-1094.

Cf. A092693, A173557, A333870.

f[p_, e_] := p - 1; u[1] = 1; u[n_] := Times @@ (f @@@ FactorInteger[n]); a[n_] := Plus @@ FixedPointList[u, n] - n - 1 ; Array[a, 100]

cp's OEIS Frontend

A343243 Sociable totient numbers of order 3: numbers k such that s(s(s(k))) = k, but s(k) != k, where s(k) = A092693(k) is the sum of iterated phi function.

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A082897 Perfect totient numbers.

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A053478 Sum of iterates when phi, A000010, is iterated until fixed point 1.

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A331273 Sum of the iterated exponential totient function (A072911).

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A092694 Product of iterated phi(n).

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A288452 Pseudoperfect totient numbers: numbers n such that equal the sum of a subset of their iterated phi(n).

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A329153 Sum of the iterated unitary totient function (A047994).

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