I searched the solutions of a16+a26+a36 = b16+b26+b36 by modp method.

I decided N=1000 because the amounts of calculation increased rapidly
when the search range was enlarged.

Example

[     67,     37,     36] = [     65,     52,     15]    2 6  means

67^n + 37^n + 36^n = 65^n + 52^n + 15^n, n=2,6


Most solutions are the case of n=2 and 6 in this search range.

Eight solutions were found as the case of n=1,2 and 6.

    a1      a2     a3        b1     b2     b3    n
[     23,     15,     10] = [     22,     19,      3]    2 6
[     67,     37,     36] = [     65,     52,     15]    2 6
[     74,     47,     33] = [     73,     54,     23]    2 6
 [     81,     43,     32] = [     80,     55,      3]    2 6 
[     81,     50,     37] = [     78,     65,     11]    2 6
[    136,    113,     51] = [    129,    125,     40]    2 6
[    138,     62,     25] = [    135,     92,     82]      6
[    147,     92,     71] = [    133,    132,      1]    2 6
[    230,    121,    111] = [    225,    169,     26]    2 6
[    245,    142,     75] = [    243,    163,     14]    2 6
[    249,    148,    103] = [    243,    188,     11]    2 6
[    267,    139,    131] = [    261,    197,     29]    2 6
[    276,    169,     65] = [    275,    179,     36]    2 6
[    293,    186,    173] = [    282,    241,    113]    2 6
[    295,    154,     75] = [    294,    173,      1]    2 6
[    300,    211,    125] = [    289,    249,     68]    2 6
[    303,   -124,   -185] = [   -300,    211,     83]  1 2 6
[    311,    173,     99] = [    309,    199,     37]    2 6
[    326,    243,    107] = [    311,    282,     23]    2 6
[    330,     55,      1] = [    311,    268,    159]      6
[    337,    277,    124] = [    317,    311,     92]    2 6
[    372,    271,    127] = [    356,    317,     27]    2 6
[    400,    149,    102] = [    398,    226,     75]      6
[   -405,    307,    140] = [   -372,    371,     43]  1 2 6
[    410,    113,   -192] = [    368,    303,   -340]  1   6
[    439,    345,    152] = [    431,    367,    120]    2 6
[    496,   -127,   -393] = [   -479,    432,     23]  1 2 6
[    508,    281,    177] = [    507,    299,    148]    2 6
[    518,    229,     34] = [    448,    424,    421]      6
[    524,    176,     53] = [    506,    395,    230]      6
[    528,    293,    271] = [    512,    409,     93]    2 6
[    537,    400,    317] = [    534,    419,    281]      6
[    556,    409,    197] = [    515,    500,      1]    2 6
[   -562,    387,    271] = [   -543,    461,    178]  1 2 6
[    612,    499,    391] = [    572,    571,    351]    2 6
[    614,    383,    201] = [    605,    445,      6]    2 6
[    625,    412,    305] = [    596,    523,    157]    2 6
[    655,    419,    415] = [    617,    569,    269]    2 6
[    710,    579,    365] = [    669,    653,    314]    2 6
[    721,    311,    152] = [    679,    575,    440]      6
[    770,    375,    353] = [    763,    501,    158]    2 6
[    771,    568,    150] = [    765,    582,    404]      6
[    775,    431,    426] = [    729,    655,     86]    2 6
[    787,    564,    298] = [    778,    604,    237]    2 6
[   -796,    627,    271] = [   -764,    699,    167]  1 2 6
[   -801,    637,    248] = [   -753,    728,    109]  1 2 6
[    865,    282,    196] = [    770,    769,    402]      6
[    885,    499,    268] = [    879,    572,     65]    2 6
[    890,    641,    443] = [    838,    781,    295]    2 6
[    891,    590,    389] = [    831,    775,     46]    2 6
[    892,    519,    359] = [    884,    603,    221]    2 6
[    919,    566,    473] = [    883,    746,    229]    2 6
[    939,    509,    508] = [    916,    711,    233]    2 6
[    957,    545,    133] = [    927,    665,    647]      6
[    960,    725,    141] = [    918,    831,    239]      6
[    983,    790,    144] = [    974,    810,    457]      6
[    991,    751,    258] = [    971,    813,     94]    2 6
[    995,    283,    110] = [    953,    778,     89]      6
[   -997,    724,    423] = [   -953,    852,    251]  1 2 6