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A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.

%I A267462 #40 Feb 16 2025 08:33:29
%S A267462 8911,1152271,10267951,14913991,64377991,67902031,139952671,178482151,
%T A267462 612816751,652969351,743404663,2000436751,2560600351,3102234751,
%U A267462 3215031751,5615659951,5883081751,7773873751,8863329511,9462932431,10501586767,11335174831,12191597551,13946829751,16157879263,21046047751
%N A267462 Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
%C A267462 Intersection of A002997 and A004215.
%C A267462 Carmichael numbers that are the sum of 4 but no fewer nonzero squares.
%C A267462 Carmichael numbers of the form 8*k + 7.
%C A267462 Subsequence of A185321.
%C A267462 Carmichael numbers of the form x^2 + y^2 + z^2 where x, y and z are integers are 561, 1105, 1729, 2465, 2821, 6601, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461, 530881, 552721, ...
%H A267462 Amiram Eldar, <a href="/A267462/b267462.txt">Table of n, a(n) for n = 1..10000</a> (calculated using data from Claude Goutier)
%H A267462 Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.
%H A267462 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 A267462 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelNumber.html">Carmichael Number</a>
%H A267462 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%e A267462 Carmichael number 561 is not a term of this sequence because 561 = 2^2 + 14^2 + 19^2.
%e A267462 Carmichael number 8911 is a term because there is no integer values of x, y and z for the equation 8911 = x^2 + y^2 + z^2.
%e A267462 Carmichael number 10585 is not a term because 10585 = 0^2 + 37^2 + 96^2.
%p A267462 filter:= proc(n)
%p A267462   local q;
%p A267462   if isprime(n) then return false fi;
%p A267462   if 2 &^ (n-1) mod n <> 1 then return false fi;
%p A267462   for q in ifactors(n)[2] do
%p A267462     if q[2] > 1 or (n-1) mod (q[1]-1) <> 0 then return false fi
%p A267462     od;
%p A267462     true
%p A267462 end proc:
%p A267462 select(filter, [seq(8*k+7, k=0..10^7)]); # _Robert Israel_, Jan 18 2016
%t A267462 Select[8*Range[1,8000000]+7, CompositeQ[#] && Divisible[#-1, CarmichaelLambda[#]] &] (* _Amiram Eldar_, Jun 26 2019 *)
%o A267462 (PARI) isA004215(n) = { my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri -7 ; if( j % 8 ==0, return(1) ) ; ) ; fouri *= 4 ; ) ; return(0) ; } { for(n=1, 400, if(isA004215(n), print1(n, ", ") ; ) ; ) ; }
%o A267462 isA002997(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 A267462 for(n=0, 1e10, if(isA002997(n) && isA004215(n), print1(n, ", ")));
%o A267462 (PARI) isA002997(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 A267462 for(n=0, 1e10, if(isA002997(k=8*n+7), print1(k, ", ")));
%Y A267462 Cf. A002997, A004215, A185321, A265237.
%K A267462 nonn
%O A267462 1,1
%A A267462 _Altug Alkan_, Jan 15 2016