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A339062 Sorted list of prime numbers in the union of 7-tuples (a,b,c,d,e,f,g) satisfying a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = a*b*c*d*e*f*g. 0

%I #12 Dec 18 2020 04:29:02

%S 2,3,5,23,37,67,181,307,359,1559,2417,59123,88327,95783,99907,304151,

%T 606839,932999,1179491,1531619,1860337,2188919,2363441,3578437,

%U 5474849,7577351,11273459,12994823,32393057,48222721,127896599,248648401,932998067,1109123111,2671715093,4488932999,9347244311

%N Sorted list of prime numbers in the union of 7-tuples (a,b,c,d,e,f,g) satisfying a^2 + b^2 + c^2 + d^2 + e^2 + f^2 + g^2 = a*b*c*d*e*f*g.

%C Prime numbers that appear in the integer solutions {X(1),X(2),...X(n)} of Markoff-Hurwitz equation X(1)^2 + ... + X(n)^2 = a*X(1)*...*X(n) for a=1 and n=7.

%C 7-tuples that are solutions of the above equation consisting only of primes appear to be very rare. In this special case the number N=X(1)*...*X(7) is equal to the sum of the squares of its prime factors (with multiplicity).

%C _Giorgos Kalogeropoulos_ has found two numbers N having 123 and 163 digits respectively.

%C The factors of the first one are {2, 2, 2, 23, 1109123111, 57766182616657495290612267717977498812931942308391, 11788844704086155814066994795339207139099517865226893357415731}, so this 7-tuple is a solution and all these primes belong to the sequence. (See Rivera's link for the second 7-tuple).

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1019.htm">Puzzle 1019. Follow-up to Puzzle 625</a>, The Prime Puzzles and Problems Connection.

%e {1, 1, 2, 2, 3, 23, 274} is a solution to the equation. So the primes 2,3,23 are terms of the sequence.

%Y Cf. A227208, A178444.

%K nonn

%O 1,1

%A _Giorgos Kalogeropoulos_, Nov 22 2020

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