A173703 Composite numbers n with the property that phi(n) divides (n-1)^2.

561, 1105, 1729, 2465, 6601, 8481, 12801, 15841, 16705, 19345, 22321, 30889, 41041, 46657, 50881, 52633, 71905, 75361, 88561, 93961, 115921, 126673, 162401, 172081, 193249, 247105, 334153, 340561, 378561, 449065, 460801, 574561, 656601, 658801, 670033

Offset

1,1

Comments

All terms are odd because if n is even, (n-1)^2 is odd and phi(n) is even for n > 2. - Donovan Johnson, Sep 08 2013

McNew showed that the number of terms in this sequence below x is O(x^(6/7)). - Tomohiro Yamada, Sep 28 2020

Links

Joerg Arndt and Donovan Johnson, Table of n, a(n) for n = 1..2000 (first 327 terms from Joerg Arndt)

José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, Integers, 12(2012), #A37

Nathan McNew, Radically weakening the Lehmer and Carmichael conditions, International Journal of Number Theory 9 (2013), 1215-1224; available from arXiv, arXiv:1210.2001 [math.NT], 2012.

Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 4, 2013.

Example

a(1) = 561 is in the sequence because 560^2 = phi(561)*980 = 320*980 = 313600.

Maple

isA173703 := proc(n)
    n <> 1 and not isprime(n) and (modp( (n-1)^2, numtheory[phi](n)) = 0 );
end proc:
for n from 1 to 10000 do
    if isA173703(n) then
        printf("%d, \n", n);
    end if;
end do: # R. J. Mathar, Nov 06 2017

Mathematica

Union[Table[If[PrimeQ[n] === False && IntegerQ[(n-1)^2/EulerPhi[n]], n], {n, 3, 100000}]]
Select[Range[700000], CompositeQ[#]&&Divisible[(#-1)^2, EulerPhi[#]]&] (* Harvey P. Dale, Nov 29 2014 *)
Select[Range[1, 700000, 2], CompositeQ[#]&&PowerMod[#-1, 2, EulerPhi[ #]] == 0&] (* Harvey P. Dale, Aug 10 2021 *)

Prog

(PARI)
N=10^9;
default(primelimit, N);
ct = 0;
{ for (n=4, N,
    if ( ! isprime(n),
        if ( ( (n-1)^2 % eulerphi(n) ) == 0,
            ct += 1;
            print(ct, " ", n);
        );
    );
); }
/* Joerg Arndt, Jun 23 2012 */

Crossrefs

Cf. A000010, A207080.

Cf. A238574 (k-Lehmer numbers for some k).

Sequence in context: A002997 A355039 A087788 * A306338 A367320 A300629

Adjacent sequences: A173700 A173701 A173702 * A173704 A173705 A173706

Keyword

nonn

Author

José María Grau Ribas, Nov 25 2010

Status

approved