cp's OEIS Frontend

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

%I A258839 #27 Apr 24 2024 02:43:46
%S A258839 561,162401,410041,488881,656601,2433601,36765901,109393201,171454321,
%T A258839 176659201,178837201,189941761,221884001,288120421,600892993,
%U A258839 618068881,721244161,931694401,985052881,1183104001,1828377001,1848112761,1943951041,2361232477,2438403661
%N A258839 Carmichael numbers whose prime factors all have the form p=1+x^2+y^2 for some x,y in Z.
%C A258839 Banks & Freiberg show that this sequence is infinite.
%H A258839 Amiram Eldar, <a href="/A258839/b258839.txt">Table of n, a(n) for n = 1..10000</a> (calculated using data from Claude Goutier; terms 1..733 from Giovanni Resta)
%H A258839 William D. Banks and Tristan Freiberg, <a href="https://doi.org/10.1016/j.jnt.2016.01.012">Carmichael numbers and the sieve</a>, Journal of Number Theory, Vol. 165 (2016), pp. 15-29; <a href="https://arxiv.org/abs/1506.03497">arXiv preprint</a>, arXiv:1506.03497 [math.NT], 2015.
%H A258839 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 A258839 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%o A258839 (PARI) has(n)=for(x=sqrtint(n\2),sqrtint(n-1), if(issquare(n-x^2-1), return(1)));0
%o A258839 Korselt(n,f=factor(n))=for(i=1,#f~,if(f[i, 2]>1||(n-1)%(f[i, 1]-1),return(0))); 1
%o A258839 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
%Y A258839 Cf. A002997 (Carmichael numbers), A079545 (primes of the form x^2 + y^2 + 1).
%K A258839 nonn
%O A258839 1,1
%A A258839 _Michel Marcus_, Jun 12 2015