
COMMENTS

Conjecture: The Diophantine equation pi(x^3) + pi(y^3) = pi(z^3) with 0 < x <= y < z has infinitely many solutions.
The 25 terms we have found yield the following 25 solutions to the equation: (x,y,z) = (3,3,4), (54,80,89), (63,85,97), (27,100,101), (47,106,110), (80,190,196), (122,223,237), (229,335,372), (151,401,410), (263,1453,1457), (1302,2382,2522), (879,3301,3327), (2190,4011,4244), (498,4434,4437), (3792,4991,5684), (4496,4584,5777), (3113,7442,7647), (5239,8090,8827), (6904,8116,9608), (5659,8910,9680), (5323,9187,9807), (5527,10168,10744), (7395,17050,17563), (11637,17438,19146), (4486,21125,21208).
See also the conjecture in A262408 involving the nth powers with n = 2,4,5,....
Solution triples (x,y,z) corresponding to a(n) for n = 26..42: (16440, 19774, 23188), (4775, 27091, 27153), (10708, 27687, 28286), (25272, 28248, 34086), (6302, 35360, 35443), (3941, 40040, 40057), (16336, 48639, 49338), (33631, 43365, 49613), (6206, 54390, 54425), (6741, 55317, 55360), (28160, 54247, 56906), (25339, 59637, 61304), (41473, 63300, 69147), (27684, 67825, 69515), (29690, 71841, 73694), (65989, 67172, 84508), (55781, 88294, 95674)  Chai Wah Wu, May 24 2018
