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%I #37 Nov 13 2022 08:01:12
%S 75,119,551,755,4501,4895,16371,56863,61091,74201,201797,336709,
%T 534793,596827,879397,1007541
%N Positive integers k such that k^2 = A^2+B^2+C^2 and A^3+B^3+C^3 = m^3, where gcd(A,B,C) = 1 and A, B, C, m are positive integers.
%C From _Chai Wah Wu_, Sep 04 2020: (Start)
%C A. Martin and R. Davis showed that 91091088729334859 = sqrt(11868013975030087^2+16269106368215226^2+88837226814909894^2) is a term (see Links).
%C Table of values for k, A, B, C, m:
%C k A B C m
%C ---------------------------------------------
%C 75 14 23 70 71
%C 119 3 34 114 115
%C 551 18 349 426 493
%C 755 145 198 714 721
%C 4501 1016 2364 3693 4013
%C 4895 213 3450 3466 4357
%C 16371 3542 9286 13009 14497
%C 56863 6213 32194 46458 51157
%C 61091 29233 29574 44754 51985
%C 74201 32913 38444 54264 63185
%C 201797 106677 117252 124876 168373
%C 336709 110051 118044 295512 306467
%C 534793 116457 286752 436136 476393
%C 596827 202023 234550 510270 536023
%C 879397 43472 613560 628485 782597
%C 1007541 272267 417416 875656 914315
%C (End)
%H A. Martin and R. Davis, <a href="https://archive.org/details/bub_gb_UuFJAQAAIAAJ/page/n225/mode/2up">Solution of problem 143</a>, Jahrbuch über die Fortschritte der Mathematik, Band 29, Jahrgang 1898, pub. 1900, p. 157.
%H Ed Pegg Jr.'s Math Puzzles, <a href="http://www.mathpuzzle.com/cbumpkin.txt">A^2 + B^2 + C^2 = Square, A^3 + B^3 + C^3 = Cube</a>
%H Seiji Tomita, <a href="http://www.maroon.dti.ne.jp/fermat/dioph196e.html">A simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} has infinitely many integer solutions</a>.
%e 56863 is in the sequence because 56863^2 = 6213^2 + 32194^2 + 46458^2, 6213^3 + 32194^3 + 46458^3 = 51157^3 and gcd(6213, 32194, 46458) = 1.
%Y Cf. A096910.
%K nonn,more
%O 1,1
%A _Mo Li_, Aug 21 2020