cp's OEIS Frontend

A207080 The smallest Carmichael number k such that phi(k) does not divide (k-1)^n, where phi is the Euler totient function.

%I A207080 #37 Apr 20 2024 07:42:24
%S A207080 561,2821,838201,41471521,45496270561,776388344641,344361421401361,
%T A207080 375097930710820681,330019822807208371201,4971170854788923506051
%N A207080 The smallest Carmichael number k such that phi(k) does not divide (k-1)^n, where phi is the Euler totient function.
%C A207080 Conjecture: phi(a(n)) divides (a(n)-1)^(n+1).
%C A207080 a(10) <= 9645020063586019926451. - _Daniel Suteu_, Dec 25 2020
%H A207080 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 A207080 José María Grau and Antonio M. Oller-Marcén, <a href="https://doi.org/10.1515/integers-2012-0025">On k-Lehmer numbers</a>, Integers, 12 (2012), #A37; <a href="http://math.colgate.edu/~integers/m37/m37.pdf">alternative link</a>; <a href="https://arxiv.org/abs/1012.2337">arXiv preprint</a>, arXiv:1012.2337 [math.NT], 2010-2012.
%H A207080 Nathan McNew, <a href="https://doi.org/10.1142/S1793042113500218">Radically weakening the Lehmer and Carmichael conditions</a>, International Journal of Number Theory, Vol. 9, No. 5 (2013), pp. 1215-1224; <a href="https://arxiv.org/abs/1210.2001">arXiv preprint</a>, arXiv:1210.2001 [math.NT], 2012.
%o A207080 (PARI) is_c(n) = { my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1; }
%o A207080 isok(k, n) = ((k-1)^n % eulerphi(k)) != 0;
%o A207080 a(n) = my(k=1); while (!(is_c(k) && isok(k,n)), k++); k; \\ _Michel Marcus_, Dec 25 2020
%Y A207080 Cf. A000010, A002997 (Carmichael numbers), A173703.
%K A207080 nonn,more
%O A207080 1,1
%A A207080 _José María Grau Ribas_, Feb 15 2012
%E A207080 a(7)-a(9) from _Richard Pinch_, Feb 18 2012
%E A207080 a(10) calculated using data from _Claude Goutier_ and added by _Amiram Eldar_, Apr 20 2024