A329417 - OEIS

%I #28 Apr 22 2024 14:28:20

%S 12876480001,102293818705,162303632569,639554081761,783962120161,

%T 3224063844001,4553777859841,10276904735461,40867660260505,

%U 51496980091921,51641004415105,52412615611201,52933062609505,73892907966241,97388953462801,107862864807061,182236335107905,210587050134721

%N Carmichael numbers m that have at least 3 prime factors p such that (p-1)*p^2 divides m-p.

%C In 1950, Giuga conjectured that there are no composite numbers n for which 1^(n-1) + 2^(n-1) + ... + (n-1)^(n-1) == -1 (mod n). If such a number exists, then it must be a Carmichael number n such that (p-1)*p^2 divides n-p for every prime p dividing n.

%D Giuseppe Giuga, Su una presumibile proprietà caratteristica dei numeri primi (in Italian), Istituto Lombardo Scienze e Lettere, Rendiconti di Classe di scienze matematiche e naturali, Vol. 83 (1950), pp. 511-528.

%H Amiram Eldar, <a href="/A329417/b329417.txt">Table of n, a(n) for n = 1..8950</a> (terms below 10^22 calculated using data from Claude Goutier; terms 1..854 from Daniel Suteu)

%H Takashi Agoh, <a href="https://doi.org/10.1007/BF02570490">On Giuga's conjecture</a>, Manuscripta Mathematica, Vol. 87, No. 1 (1995), pp. 501-510.

%H William D. Banks, C. Wesley Nevans and Carl Pomerance, <a href="http://albanian-j-math.com/archives/2009-11.pdf">A remark on Giuga's conjecture and Lehmer's totient problem</a>, Albanian Journal of Mathematics, Vol. 3, No. 2 (2009), pp. 81-85; <a href="https://math.dartmouth.edu/~carlp/Carmichael_giuga.pdf">alternative link</a>.

%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GiugasConjecture.html">Giuga's Conjecture</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Agoh%E2%80%93Giuga_conjecture">Agoh-Giuga conjecture</a>.

%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.

%e m = 12876480001 is a term because it is a Carmichael number, and it has at least 3 prime factors p, {7, 11, 37}, such that (p-1)*p^2 divides m-p.

%o (Perl) use bigint; use ntheory ':all'; sub isok { my $m = $_[0]; is_carmichael($m) && (grep { ($m-$_) % (($_-1)*$_*$_) == 0 } factor($m)) >= 3 };

%Y Cf. A002997.

%K nonn

%O 1,1

%A _Amiram Eldar_ and _Daniel Suteu_, Nov 29 2019