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A337251 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.

%I A337251 #37 Nov 13 2022 08:01:12
%S A337251 75,119,551,755,4501,4895,16371,56863,61091,74201,201797,336709,
%T A337251 534793,596827,879397,1007541
%N A337251 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 A337251 From _Chai Wah Wu_, Sep 04 2020: (Start)
%C A337251 A. Martin and R. Davis showed that 91091088729334859 = sqrt(11868013975030087^2+16269106368215226^2+88837226814909894^2) is a term (see Links).
%C A337251 Table of values for k, A, B, C, m:
%C A337251   k        A        B        C        m
%C A337251   ---------------------------------------------
%C A337251   75       14       23       70       71
%C A337251   119      3        34       114      115
%C A337251   551      18       349      426      493
%C A337251   755      145      198      714      721
%C A337251   4501     1016     2364     3693     4013
%C A337251   4895     213      3450     3466     4357
%C A337251   16371    3542     9286     13009    14497
%C A337251   56863    6213     32194    46458    51157
%C A337251   61091    29233    29574    44754    51985
%C A337251   74201    32913    38444    54264    63185
%C A337251   201797   106677   117252   124876   168373
%C A337251   336709   110051   118044   295512   306467
%C A337251   534793   116457   286752   436136   476393
%C A337251   596827   202023   234550   510270   536023
%C A337251   879397   43472    613560   628485   782597
%C A337251   1007541  272267   417416   875656   914315
%C A337251 (End)
%H A337251 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 A337251 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 A337251 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 A337251 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 A337251 Cf. A096910.
%K A337251 nonn,more
%O A337251 1,1
%A A337251 _Mo Li_, Aug 21 2020