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

1,1

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.

Prog

(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

Crossrefs

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

Sequence in context: A329460 A097061 A290497 * A371759 A213867 A139089

Adjacent sequences: A258836 A258837 A258838 * A258840 A258841 A258842

Keyword

nonn

Author

Michel Marcus, Jun 12 2015

Status

approved