Equal Sums of Like Powers $(11.1.n)$ for $10\le n \le 19$.

Suppose $a(n)$ is the minimum integer $k$ such that $k^{11}$ can be expressed as the sum of $n$ distinct positive 11th powers.

Q: Find $a(n)$ for $10\le n\le 19$.

For example, $a(20)=119$ because $199^{11}$ is the sum of 20 terms of $11$th powers,

$119^{11}=118^{11}+90^{11}+81^{11}+80^{11}+76^{11}+72^{11}+69^{11}+63^{11}+56^{11}+53^{11}+52^{11}+46^{11}+41^{11}+38^{11}+35^{11}+26^{11}+23^{11}+13^{11}+6^{11}+5^{11}$

$a(121)=204$ because $204^{11}$ is the sum of 121 terms of $11$th powers,

$204^{11}=161^{11}+160^{11}+159^{11}+158^{11}+157^{11}+156^{11}+155^{11}+154^{11}+153^{11}+152^{11}+151^{11}+150^{11}+149^{11}+147^{11}+146^{11}+145^{11}+144^{11}+143^{11}+142^{11}+141^{11}+140^{11}+139^{11}+138^{11}+137^{11}+136^{11}+135^{11}+134^{11}+133^{11}+132^{11}+131^{11}+130^{11}+129^{11}+128^{11}+127^{11}+126^{11}+125^{11}+124^{11}+123^{11}+122^{11}+121^{11}+120^{11}+119^{11}+118^{11}+117^{11}+116^{11}+115^{11}+114^{11}+113^{11}+112^{11}+111^{11}+110^{11}+109^{11}+108^{11}+107^{11}+106^{11}+104^{11}+103^{11}+102^{11}+101^{11}+100^{11}+99^{11}+98^{11}+97^{11}+96^{11}+95^{11}+94^{11}+93^{11}+92^{11}+91^{11}+90^{11}+89^{11}+88^{11}+87^{11}+86^{11}+84^{11}+83^{11}+82^{11}+81^{11}+80^{11}+79^{11}+78^{11}+77^{11}+76^{11}+75^{11}+74^{11}+73^{11}+71^{11}+70^{11}+69^{11}+68^{11}+67^{11}+66^{11}+65^{11}+60^{11}+59^{11}+49^{11}+44^{11}+41^{11}+39^{11}+31^{11}+28^{11}+27^{11}+25^{11}+23^{11}+22^{11}+21^{11}+18^{11}+17^{11}+16^{11}+14^{11}+13^{11}+11^{11}+10^{11}+9^{11}+8^{11}+7^{11}+6^{11}+5^{11}+3^{11}+2^{11}+1^{11}$

I have found all a(k) for $20\le n\le 101$.

Current progress：

a(10) >= 285
a(11) >= 232
a(12) >= 198
a(13) >= 171
a(14) >= 153
a(15) >= 301
a(16) >= 201
a(17) >= 133
a(18) >= 123
a(19) >= 116

See also: Equal Sums of Like Powers 11.1.n (An academic online discussion forum for the Mathematics Research Network)