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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 (list; graph; refs; listen; history; text; internal format)
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

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

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

Cf. A002997, A004431, A122045.

Sequence in context: A102924 A214017 A083738 * A291602 A275881 A291616

Adjacent sequences:  A265234 A265235 A265236 * A265238 A265239 A265240

KEYWORD

nonn

AUTHOR

Altug Alkan, Dec 06 2015

STATUS

approved

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)