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

561, 2821, 838201, 41471521, 45496270561, 776388344641, 344361421401361, 375097930710820681, 330019822807208371201, 4971170854788923506051

Offset: 1

Author

José María Grau Ribas, Feb 15 2012

Keywords

• nonn
• more

Comments

Conjecture: phi(a(n)) divides (a(n)-1)^(n+1).

a(10) <= 9645020063586019926451. - Daniel Suteu, Dec 25 2020

Links

• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• José María Grau and Antonio M. Oller-Marcén, On k-Lehmer numbers, Integers, 12 (2012), #A37; alternative link; arXiv preprint, arXiv:1012.2337 [math.NT], 2010-2012.
• Nathan McNew, Radically weakening the Lehmer and Carmichael conditions, International Journal of Number Theory, Vol. 9, No. 5 (2013), pp. 1215-1224; arXiv preprint, arXiv:1210.2001 [math.NT], 2012.

Crossrefs

Cf. A000010, A002997 (Carmichael numbers), A173703.

Programs

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; }
isok(k, n) = ((k-1)^n % eulerphi(k)) != 0;
a(n) = my(k=1); while (!(is_c(k) && isok(k,n)), k++); k; \\ Michel Marcus, Dec 25 2020

Extensions

a(7)-a(9) from Richard Pinch, Feb 18 2012

a(10) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024