A258839 Carmichael numbers whose prime factors all have the form p=1+x^2+y^2 for some x,y in Z.

561, 162401, 410041, 488881, 656601, 2433601, 36765901, 109393201, 171454321, 176659201, 178837201, 189941761, 221884001, 288120421, 600892993, 618068881, 721244161, 931694401, 985052881, 1183104001, 1828377001, 1848112761, 1943951041, 2361232477, 2438403661

Offset: 1

Author

Michel Marcus, Jun 12 2015

Keywords

• nonn

Comments

Banks & Freiberg show that this sequence is infinite.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..733 from Giovanni Resta)
• William D. Banks and Tristan Freiberg, Carmichael numbers and the sieve, Journal of Number Theory, Vol. 165 (2016), pp. 15-29; arXiv preprint, arXiv:1506.03497 [math.NT], 2015.
• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Index entries for sequences related to Carmichael numbers.

Crossrefs

Cf. A002997 (Carmichael numbers), A079545 (primes of the form x^2 + y^2 + 1).

Programs

PARI

has(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(1)));0
Korselt(n,f=factor(n))=for(i=1,#f~,if(f[i, 2]>1||(n-1)%(f[i, 1]-1),return(0))); 1
is(n)=my(f); if(n%2==0||isprime(n)||!Korselt(n,f=factor(n))||n<9, return(0)); for(i=1,#f~,if(!has(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Jun 12 2015