A265237 Carmichael numbers (A002997) that are the sum of two squares.

1105, 2465, 10585, 29341, 46657, 115921, 162401, 252601, 278545, 294409, 314821, 410041, 488881, 530881, 552721, 1461241, 1909001, 2433601, 3224065, 3581761, 4335241, 5148001, 5310721, 5444489, 5632705, 6054985, 6189121, 7207201, 7519441, 8134561, 8355841

Offset: 1

Author

Altug Alkan, Dec 06 2015

Keywords

• nonn

Comments

Carmichael numbers that are the sum of two distinct nonzero squares.

29341 is the first term for which neither of the squares can be the square of a prime.

Carmichael numbers that are not the sum of two squares start 561, 1729, 2821, 6601, 8911, 15841, ...

A Carmichael number m is a sum of two squares if and only if p == 1 (mod m) for every prime p|m. Observation, numerically checked by Amiram Eldar: the first 13 terms of this sequence are odd composites m such that m | EulerNumber(m-1) (A122045). - Thomas Ordowski, Mar 01 2020

Examples

			1105 is a term because 1105 = 23^2 + 24^2.
2465 is a term because 2465 = 41^2 + 28^2.
10585 is a term because 10585 = 37^2 + 96^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
• Eric Weisstein's World of Mathematics, Carmichael Number
• Index entries for sequences related to Carmichael numbers

Crossrefs

Cf. A002997, A004431, A122045.

Programs

Mathematica

t = Cases[Range[1, 10^7, 2], n_ /; Mod[n, CarmichaelLambda@ n] == 1 && ! PrimeQ@ n]; Select[t, SquaresR[2, #] > 0 &] (* Michael De Vlieger, Dec 06 2015, after Artur Jasinski at A002997 *)

PARI

is(n)=if(n<5, return(0)); my(f=factor(n)%4); if(vecmin(f[, 1])>1, return(0)); for(i=1, #f[, 1], if(f[i, 1]==3 && f[i, 2]%2, return(0))); 1
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}
for(n=1, 1e7, if(is(n)&&is_c(n), print1(n, ", ")))