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A267462
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Carmichael numbers that are not of the form x^2 + y^2 + z^2 where x, y and z are integers.
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1
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8911, 1152271, 10267951, 14913991, 64377991, 67902031, 139952671, 178482151, 612816751, 652969351, 743404663, 2000436751, 2560600351, 3102234751, 3215031751, 5615659951, 5883081751, 7773873751, 8863329511, 9462932431, 10501586767, 11335174831, 12191597551, 13946829751, 16157879263, 21046047751
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OFFSET
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1,1
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COMMENTS
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Carmichael numbers that are the sum of 4 but no fewer nonzero squares.
Carmichael numbers of the form 8*k + 7.
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, ...
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LINKS
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G. Tarry, I. Franel, A. Korselt, and G. Vacca. Problème chinois. L'intermédiaire des mathématiciens 6 (1899), pp. 142-144.
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EXAMPLE
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Carmichael number 561 is not a term of this sequence because 561 = 2^2 + 14^2 + 19^2.
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.
Carmichael number 10585 is not a term because 10585 = 0^2 + 37^2 + 96^2.
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MAPLE
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filter:= proc(n)
local q;
if isprime(n) then return false fi;
if 2 &^ (n-1) mod n <> 1 then return false fi;
for q in ifactors(n)[2] do
if q[2] > 1 or (n-1) mod (q[1]-1) <> 0 then return false fi
od;
true
end proc:
select(filter, [seq(8*k+7, k=0..10^7)]); # Robert Israel, Jan 18 2016
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MATHEMATICA
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Select[8*Range[1, 8000000]+7, CompositeQ[#] && Divisible[#-1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 26 2019 *)
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PROG
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(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, ", ") ; ) ; ) ; }
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 }
for(n=0, 1e10, if(isA002997(n) && isA004215(n), print1(n, ", ")));
(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 }
for(n=0, 1e10, if(isA002997(k=8*n+7), print1(k, ", ")));
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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