The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A082897

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A082897

Perfect totient numbers.
(history; published version)

#92 by Michel Marcus at Fri Mar 24 12:43:21 EDT 2023

STATUS

reviewed

approved

#91 by Joerg Arndt at Fri Mar 24 12:05:31 EDT 2023

STATUS

proposed

reviewed

#90 by Chai Wah Wu at Fri Mar 24 11:41:12 EDT 2023

STATUS

editing

proposed

#89 by Chai Wah Wu at Fri Mar 24 11:40:58 EDT 2023

PROG

(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

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

STATUS

approved

editing

#88 by Bruno Berselli at Wed Mar 10 03:18:54 EST 2021

STATUS

reviewed

approved

#87 by Michel Marcus at Wed Mar 10 03:11:28 EST 2021

STATUS

proposed

reviewed

#86 by Amiram Eldar at Wed Mar 10 02:24:46 EST 2021

STATUS

editing

proposed

#85 by Amiram Eldar at Wed Mar 10 01:48:30 EST 2021

REFERENCES

M. V. Subbarao, On a Function connected with phi(n), The Mathematics Student, Vol. 23 (1955), pp. 178-179.

#84 by Amiram Eldar at Wed Mar 10 01:47:14 EST 2021

COMMENTS

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

LINKS

Jovele G. Belmonte, <a href="https://animorepository.dlsu.edu.ph/etd_masteral/3449/">On perfect totient numbers</a>, Masteral Thesis, De La Salle University, 2006.

Li-Xia Dai and Yong-Gao Chen, <a href="https://caod.oriprobe.com/articles/13427719/A_note_on_perfect_totient_numbers.htm">A note on perfect totient numbers</a>, Journal of Northeast Normal University, Vol. 39, No. 4 (2007), pp. 17-19.

Tuukka Hyvärinen, <a href="https://trepo.tuni.fi/handle/10024/97744">Täydelliset totienttiluvut</a> (in Finnish), Master's thesis, Tampere University, 2015; <a href="https://core.ac.uk/download/pdf/250138377.pdf">alternative link</a>.

Florian Luca, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Luca/luca66.html">On the Distribution of Perfect Totients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.4.

Hans Sieburg and Michael Kentgens, <a href="https://www.researchgate.net/publication/258424386_On_Phi-perfect_numbers">On Phi-perfect numbers</a>, in: J. Akiyama et al. (eds.), Number Theory and Combinatorics, Japan 1984, World Scientific, 1985, pp. 245-254.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Perfect_totient_number">Perfect totient number</a>.

#83 by Amiram Eldar at Wed Mar 10 01:43:39 EST 2021

REFERENCES

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B41, pp. 147-150.

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).

T. Venkataraman, Perfect totient number, The Mathematics Student, Vol. 43 (1975), p. 178.

LINKS

Douglas E. Iannucci, Deng Moujie and Graeme L. Cohen, <a href="https://www.cs.uwaterloo.ca/journals/JIS/VOL6/Cohen2/cohen50.html">On Perfect Totient Numbers</a>, J. Journal of Integer Sequences, Vol. 6 (2003), #Article 03.4.5.

D. Deng Moujie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Deng/deng1.html">A Note On Perfect Totient Numbers</a>, JIS Journal of Integer Sequences, Vol. 12 (2009) #, Article 09.6.2.