A092694 -id:A092694 - cp's OEIS frontend

A349306 Numbers k that divide A092694(k).

1, 8, 16, 27, 32, 54, 64, 81, 108, 125, 128, 162, 216, 243, 250, 256, 324, 343, 432, 486, 500, 512, 625, 648, 686, 729, 864, 972, 1000, 1024, 1029, 1250, 1296, 1331, 1372, 1458, 1728, 1944, 2000, 2048, 2058, 2187, 2197, 2401, 2500, 2592, 2662, 2744, 2916, 3087

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Includes all the numbers of the form 2^k * m, where k >= 0 and m is an odd cubefull number, except for 2 and 4. In particular, includes all the cubefull numbers (A036966).

Terms whose odd part is not cubefull are 1029, 2058, 3087, 4116, 6174, 6591, ...

			8 is a term since A092694(8) = 8 is divisible by 8.
16 is a term since A092694(16) = 64 is divisible by 16.

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

Cf. A000265, A082897, A092694.

A036966 is a subsequence.

Select[Range[3000], Divisible[Times @@ FixedPointList[EulerPhi, #]/#, #] &]

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

A291782 Let f_k(n) be the result of applying phi (the Euler totient function A000010) k times to n; a(n) = n*Product_{k=1..oo} f_k(n).

Original entry on oeis.org

1, 2, 6, 8, 40, 12, 84, 64, 108, 80, 880, 96, 1248, 168, 960, 1024, 17408, 216, 4104, 1280, 2016, 1760, 40480, 1536, 32000, 2496, 5832, 2688, 77952, 1920, 59520, 32768, 42240, 34816, 53760, 3456, 127872, 8208, 59904, 40960, 1679360, 4032, 173376, 56320

Offset: 1

N. J. A. Sloane, Sep 02 2017

The logarithmic scatterplot of this sequence shows a banded structure similar to that of A092694. - Rémy Sigrist, Sep 03 2017

			Under phi, 7 -> 6 -> 2 -> 1, so a(7) = 7*6*2 = 84.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

Cf. A000010, A092694, A291783.

Table[Times @@ FixedPointList[EulerPhi, n], {n, 44}] (* Michael De Vlieger, Sep 03 2017 *)

a(n) = n * A092694(n). - Rémy Sigrist, Sep 03 2017

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A349306 Numbers k that divide A092694(k).

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A092693 Sum of iterated phi(n).

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A291782 Let f_k(n) be the result of applying phi (the Euler totient function A000010) k times to n; a(n) = n*Product_{k=1..oo} f_k(n).

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