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A265237
Carmichael numbers (A002997) that are the sum of two squares.
4
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
1,1
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
LINKS
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
EXAMPLE
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.
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 *)
PROG
(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, ", ")))
CROSSREFS
KEYWORD
nonn
AUTHOR
Altug Alkan, Dec 06 2015
STATUS
approved