cp's OEIS Frontend

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

%I A265237 #45 Feb 16 2025 08:33:27
%S A265237 1105,2465,10585,29341,46657,115921,162401,252601,278545,294409,
%T A265237 314821,410041,488881,530881,552721,1461241,1909001,2433601,3224065,
%U A265237 3581761,4335241,5148001,5310721,5444489,5632705,6054985,6189121,7207201,7519441,8134561,8355841
%N A265237 Carmichael numbers (A002997) that are the sum of two squares.
%C A265237 Carmichael numbers that are the sum of two distinct nonzero squares.
%C A265237 29341 is the first term for which neither of the squares can be the square of a prime.
%C A265237 Carmichael numbers that are not the sum of two squares start 561, 1729, 2821, 6601, 8911, 15841, ...
%C A265237 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
%H A265237 Amiram Eldar, <a href="/A265237/b265237.txt">Table of n, a(n) for n = 1..10000</a>
%H A265237 G. Tarry, I. Franel, A. Korselt, and G. Vacca. <a href="https://oeis.org/wiki/File:Probl%C3%A8me_chinois.pdf">Problème chinois</a>. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
%H A265237 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%H A265237 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>
%e A265237 1105 is a term because 1105 = 23^2 + 24^2.
%e A265237 2465 is a term because 2465 = 41^2 + 28^2.
%e A265237 10585 is a term because 10585 = 37^2 + 96^2.
%t A265237 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 *)
%o A265237 (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
%o A265237 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}
%o A265237 for(n=1, 1e7, if(is(n)&&is_c(n), print1(n, ", ")))
%Y A265237 Cf. A002997, A004431, A122045.
%K A265237 nonn
%O A265237 1,1
%A A265237 _Altug Alkan_, Dec 06 2015