A177006 -id:A177006 - cp's OEIS frontend

A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

1, 2, 8, 36, 128, 225, 289, 578, 900, 2025, 2601, 3600, 10404, 32768, 41616, 45369, 57600, 242064, 665856, 725904, 783225, 1134225, 1140624, 1782225, 1988100, 2903616, 3132900, 4862025, 6155361, 6275025, 7128900, 7868025, 8625969, 10208025, 13505625

Offset: 1

Farideh Firoozbakht, Dec 12 2014

nonn

2^m is a term of the sequence if and only if m=2^j-1 where j is a nonnegative integer. Hence the sequence is infinite.
289 is a term of the sequence which is of the form p^2 where p is prime. What is the next such term?
578 is a term of the sequence which is not of the form 2^m or m^2. What is the next such term?
A248862 gives primes p such that 900*p^2 is a term of the sequence.
Subsequence of A055008. - Jason Yuen, Jul 01 2024

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A177006, A248862, A055008.

Prepend[Select[Range[30000], Mod[EulerPhi[#]^EulerPhi[#], DivisorSigma[1, #]] == 1 &], 1] (* Michael De Vlieger, Dec 13 2014 *)

isok(n) = my(in = eulerphi(n)); lift(Mod(in, sigma(n))^in - 1) == 0; \\ Michel Marcus, Dec 13 2014

A177012 Numbers k such that k^k == -1 (mod phi(k)).

Original entry on oeis.org

1, 2, 3, 15, 87, 255, 11759, 26279, 39455, 43919, 65535, 112895, 443807, 1347455, 1464911, 1568255, 1604559, 1968095, 2441559, 5948799, 16210655, 39624767, 39839039, 59187455, 81624279, 83623935, 251009695, 256685439, 338979839, 434357967, 455345855, 471783935, 487722815, 518291135, 596835839

Offset: 1

Farideh Firoozbakht, May 19 2010

nonn

3 is the largest prime term of this sequence.
All terms are squarefree. There is no further term up to 2*10^8.
If phi(k) divides k+1 then k is in the sequence. This implies A050474 and A203966 are subsequences of this sequence. - Jahangeer Kholdi, Dec 10 2014

			phi(15)=8 and 15^15 == -1 (mod 8), so 15 is in the sequence.

Cf. A000010, A177006.

Cf. A050474, A203966.

v={};Do[If[PowerMod[n,n,EulerPhi[n]]==EulerPhi[n]-1,AppendTo[v,n];
Print[v]],{n,200000000}]

a(27)-a(29) from Jahangeer Kholdi, Dec 10 2014

a(30)-a(35) from Farideh Firoozbakht, Dec 10 2014

A181476 Numbers k such that k^k == 1 (mod sigma(k)).

Original entry on oeis.org

1, 2, 8, 9, 36, 128, 576, 625, 900, 1156, 2304, 2601, 5185, 6561, 10082, 10404, 27225, 32768, 57600, 117649, 181476, 260100, 285156, 367236, 378225, 443521, 607825, 617796, 645248, 656100, 665856, 783225, 1115136, 1394450, 1500625, 1782225

Offset: 1

Farideh Firoozbakht, May 23 2010

nonn

If p is prime then p^m is in the sequence iff m is of the form p^t-1 where t is a nonnegative integer.

Cf. A000203, A177006.

Join[{1}, Select[Range[1800000], PowerMod[ #,#,DivisorSigma[1,# ]]==1 &]]

isok(k) = Mod(k, sigma(k))^k == 1; \\ Michel Marcus, Feb 09 2021

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A248861 Numbers k such that phi(k)^phi(k) == 1 (mod sigma(k)).

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A177012 Numbers k such that k^k == -1 (mod phi(k)).

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A181476 Numbers k such that k^k == 1 (mod sigma(k)).

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