A329417 Carmichael numbers m that have at least 3 prime factors p such that (p-1)*p^2 divides m-p.

12876480001, 102293818705, 162303632569, 639554081761, 783962120161, 3224063844001, 4553777859841, 10276904735461, 40867660260505, 51496980091921, 51641004415105, 52412615611201, 52933062609505, 73892907966241, 97388953462801, 107862864807061, 182236335107905, 210587050134721

Offset: 1

Author

Amiram Eldar and Daniel Suteu, Nov 29 2019

Keywords

• nonn

Comments

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.

Examples

			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.
		

References

• 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.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..8950 (terms below 10^22 calculated using data from Claude Goutier; terms 1..854 from Daniel Suteu)
• Takashi Agoh, On Giuga's conjecture, Manuscripta Mathematica, Vol. 87, No. 1 (1995), pp. 501-510.
• William D. Banks, C. Wesley Nevans and Carl Pomerance, A remark on Giuga's conjecture and Lehmer's totient problem, Albanian Journal of Mathematics, Vol. 3, No. 2 (2009), pp. 81-85; alternative link.
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Eric Weisstein's World of Mathematics, Giuga's Conjecture.
• Wikipedia, Agoh-Giuga conjecture.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A002997.

Programs

Perl

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