The lowest K with palindromic solutions is 1512. Lower K values have not been ruled out for non-palindromic solutions.

41+251+2+268+169+50+169+268+2+251+41 == 1512
42+244+23+232+211+8+211+232+23+244+42 == 1512
42+245+15+260+155+78+155+260+15+245+42 == 1512
42+246+7+288+99+148+99+288+7+246+42 == 1512
43+238+36+224+197+36+197+224+36+238+43 == 1512
43+239+28+252+141+106+141+252+28+239+43 == 1512
43+240+20+280+85+176+85+280+20+240+43 == 1512
43+241+12+308+29+246+29+308+12+241+43 == 1512
44+232+49+216+183+64+183+216+49+232+44 == 1512
44+233+41+244+127+134+127+244+41+233+44 == 1512
44+234+33+272+71+204+71+272+33+234+44 == 1512
44+235+25+300+15+274+15+300+25+235+44 == 1512
45+225+70+180+225+22+225+180+70+225+45 == 1512
45+226+62+208+169+92+169+208+62+226+45 == 1512
45+227+54+236+113+162+113+236+54+227+45 == 1512
45+228+46+264+57+232+57+264+46+228+45 == 1512
45+229+38+292+1+302+1+292+38+229+45 == 1512
46+219+83+172+211+50+211+172+83+219+46 == 1512
46+220+75+200+155+120+155+200+75+220+46 == 1512
46+221+67+228+99+190+99+228+67+221+46 == 1512
46+222+59+256+43+260+43+256+59+222+46 == 1512
47+212+104+136+253+8+253+136+104+212+47 == 1512
47+213+96+164+197+78+197+164+96+213+47 == 1512
47+214+88+192+141+148+141+192+88+214+47 == 1512
47+215+80+220+85+218+85+220+80+215+47 == 1512
47+216+72+248+29+288+29+248+72+216+47 == 1512
48+206+117+128+239+36+239+128+117+206+48 == 1512
48+207+109+156+183+106+183+156+109+207+48 == 1512
48+208+101+184+127+176+127+184+101+208+48 == 1512
48+209+93+212+71+246+71+212+93+209+48 == 1512
48+210+85+240+15+316+15+240+85+210+48 == 1512
49+200+130+120+225+64+225+120+130+200+49 == 1512
49+201+122+148+169+134+169+148+122+201+49 == 1512
49+202+114+176+113+204+113+176+114+202+49 == 1512
49+203+106+204+57+274+57+204+106+203+49 == 1512
49+204+98+232+1+344+1+232+98+204+49 == 1512
50+193+151+84+267+22+267+84+151+193+50 == 1512
50+194+143+112+211+92+211+112+143+194+50 == 1512
50+195+135+140+155+162+155+140+135+195+50 == 1512
50+196+127+168+99+232+99+168+127+196+50 == 1512
50+197+119+196+43+302+43+196+119+197+50 == 1512
51+187+164+76+253+50+253+76+164+187+51 == 1512
51+188+156+104+197+120+197+104+156+188+51 == 1512
51+189+148+132+141+190+141+132+148+189+51 == 1512
51+190+140+160+85+260+85+160+140+190+51 == 1512
51+191+132+188+29+330+29+188+132+191+51 == 1512
52+180+185+40+295+8+295+40+185+180+52 == 1512
52+181+177+68+239+78+239+68+177+181+52 == 1512
52+182+169+96+183+148+183+96+169+182+52 == 1512
52+183+161+124+127+218+127+124+161+183+52 == 1512
52+184+153+152+71+288+71+152+153+184+52 == 1512
52+185+145+180+15+358+15+180+145+185+52 == 1512
53+174+198+32+281+36+281+32+198+174+53 == 1512
53+175+190+60+225+106+225+60+190+175+53 == 1512
53+176+182+88+169+176+169+88+182+176+53 == 1512
53+177+174+116+113+246+113+116+174+177+53 == 1512
53+178+166+144+57+316+57+144+166+178+53 == 1512
53+179+158+172+1+386+1+172+158+179+53 == 1512
54+168+211+24+267+64+267+24+211+168+54 == 1512
54+169+203+52+211+134+211+52+203+169+54 == 1512
54+170+195+80+155+204+155+80+195+170+54 == 1512
54+171+187+108+99+274+99+108+187+171+54 == 1512
54+172+179+136+43+344+43+136+179+172+54 == 1512
55+162+224+16+253+92+253+16+224+162+55 == 1512
55+163+216+44+197+162+197+44+216+163+55 == 1512
55+164+208+72+141+232+141+72+208+164+55 == 1512
55+165+200+100+85+302+85+100+200+165+55 == 1512
55+166+192+128+29+372+29+128+192+166+55 == 1512
56+156+237+8+239+120+239+8+237+156+56 == 1512
56+157+229+36+183+190+183+36+229+157+56 == 1512
56+158+221+64+127+260+127+64+221+158+56 == 1512
56+159+213+92+71+330+71+92+213+159+56 == 1512
56+160+205+120+15+400+15+120+205+160+56 == 1512
57+150+250+0+225+148+225+0+250+150+57 == 1512
57+151+242+28+169+218+169+28+242+151+57 == 1512
57+152+234+56+113+288+113+56+234+152+57 == 1512
57+153+226+84+57+358+57+84+226+153+57 == 1512
57+154+218+112+1+428+1+112+218+154+57 == 1512
58+145+255+20+155+246+155+20+255+145+58 == 1512
58+146+247+48+99+316+99+48+247+146+58 == 1512
58+147+239+76+43+386+43+76+239+147+58 == 1512
59+139+268+12+141+274+141+12+268+139+59 == 1512
59+140+260+40+85+344+85+40+260+140+59 == 1512
59+141+252+68+29+414+29+68+252+141+59 == 1512
60+133+281+4+127+302+127+4+281+133+60 == 1512
60+134+273+32+71+372+71+32+273+134+60 == 1512
60+135+265+60+15+442+15+60+265+135+60 == 1512
61+128+286+24+57+400+57+24+286+128+61 == 1512
61+129+278+52+1+470+1+52+278+129+61 == 1512
62+122+299+16+43+428+43+16+299+122+62 == 1512
63+116+312+8+29+456+29+8+312+116+63 == 1512
64+110+325+0+15+484+15+0+325+110+64 == 1512