A258839 - OEIS

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

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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