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.

2, 3, 5, 23, 37, 67, 181, 307, 359, 1559, 2417, 59123, 88327, 95783, 99907, 304151, 606839, 932999, 1179491, 1531619, 1860337, 2188919, 2363441, 3578437, 5474849, 7577351, 11273459, 12994823, 32393057, 48222721, 127896599, 248648401, 932998067, 1109123111, 2671715093, 4488932999, 9347244311

Offset

1,1

Comments

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.

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).

Giorgos Kalogeropoulos has found two numbers N having 123 and 163 digits respectively.

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).

Links

Table of n, a(n) for n=1..37.

Carlos Rivera, Puzzle 1019. Follow-up to Puzzle 625, The Prime Puzzles and Problems Connection.

Example

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

Crossrefs

Cf. A227208, A178444.

Sequence in context: A138170 A105885 A228830 * A215313 A215317 A104736

Adjacent sequences: A339059 A339060 A339061 * A339063 A339064 A339065

Keyword

nonn

Author

Giorgos Kalogeropoulos, Nov 22 2020

Status

approved