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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.

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`%I #12 Dec 18 2020 04:29:02
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`%S 2,3,5,23,37,67,181,307,359,1559,2417,59123,88327,95783,99907,304151,
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`%T 606839,932999,1179491,1531619,1860337,2188919,2363441,3578437,
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`%U 5474849,7577351,11273459,12994823,32393057,48222721,127896599,248648401,932998067,1109123111,2671715093,4488932999,9347244311
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`%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.
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`%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.
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`%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).
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`%C _Giorgos Kalogeropoulos_ has found two numbers N having 123 and 163 digits respectively.
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`%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).
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`%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.
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`%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.
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`%Y Cf. A227208, A178444.
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`%K nonn
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`%O 1,1
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`%A _Giorgos Kalogeropoulos_, Nov 22 2020
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