六次等幂和
对于六次等幂和 \(\D k_{min}^6=\sum_{i=1}^n{a_i^6} \),其中 \(a_i\) 互不相等\(n=87\) 时, \(k_{min}=147\) ：
\(147^6=108^6+96^6+95^6+93^6+92^6+90^6+89^6+88^6+87^6+86^6+84^6+81^6+80^6+79^6+78^6+76^6+75^6+74^6+73^6+72^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+63^6+62^6+61^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+48^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+26^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6\)
\(n=89\) 时, \(k_{min}=150\) ：
\(150^6=110^6+101^6+97^6+95^6+93^6+91^6+89^6+88^6+87^6+86^6+85^6+83^6+82^6+81^6+79^6+78^6+77^6+76^6+74^6+73^6+72^6+71^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+63^6+62^6+61^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+49^6+48^6+47^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+27^6+26^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6\)
现在的问题是：\(n=88\) 时, \(k_{min}=?\) 穷举的 \ A393280
NAME
Minimum k such that k^6 can be expressed as the sum of n distinct positive 6th powers.
DATA
1141, 251, 123, 54, 53, 46, 43, 42, 35, 25, 32, 37, 41, 48, 49, 49, 49, 48, 45, 47, 48, 52, 55, 55, 53, 50, 55, 59, 57, 65, 67, 63, 66, 65, 68, 67, 71, 73, 77, 79, 79, 81, 83, 81, 83, 85, 89, 90, 90, 93, 94, 94, 96, 100, 121, 104, 104, 105, 103, 105, 108, 111
OFFSET
7,1
LINKS
Zhining Yang, <a href="/A393280/b393280_1.txt">Table of n, a(n) for n = 7..87</a>
Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation6thPowers.html">Diophantine Equation--6th Powers</a>.
EXAMPLE
a(7) = 1141 because 1141^6 = 1077^6 + 894^6 + 702^6 + 474^6 + 402^6 + 234^6 + 74^6 and no integer smaller than 1141 can be expressed as the sum of 7 distinct positive 6th powers.
a(9) = 123 because 123^6 = 112^6 + 106^6 + 62^6 + 46^6 + 40^6 + 22^6 + 8^6 + 5^6 + 4^6 and no integer smaller than 123 can be expressed as the sum of 9 distinct positive 6th powers.
MATHEMATICA
a:=FirstCase, k_/; Length^6], DuplicateFreeQ]]>0]; a
CROSSREFS
Cf. A130012, A130022, A018935, A252476, A252486, A261572, A394979.
对于 n<=100,目前只有n=88未搜到。 本帖最后由 northwolves 于 2026-4-14 18:16 编辑
Find n such that n^r = sum of k distinct positive r-th powerswhere k = 3-100.
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Enter the range of n (two integers, e.g. 1 20): 145 150
Enter the number of terms k (3-100): 88
Enter the number of power r (3,4,5,6,7,8,9,10,11): 6
Searching (k = 88) ...
Progress:66%[=================================> ]Checking n = 148 你的代码是怎么搜索的？ 不过6.1.7数值这么大，不知道wolfram上是怎么搜索出来的。
我试着写了个代码，现在运行18分钟，还有n=100,95,86,87,88,89,6,7,8没有找到结果。不过根据wolfram, 6应该是没有结果，而7估计我的代码找不到，8也有些困难。 7,8直接采用了Diophantine Equation--6th Powers的数据 本帖最后由 northwolves 于 2026-4-14 22:24 编辑
$n=86, 145^6=106^6+105^6+98^6+91^6+90^6+87^6+86^6+85^6+84^6+81^6+80^6+78^6+77^6+76^6+75^6+74^6+72^6+71^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+63^6+62^6+61^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+52^6+51^6+50^6+49^6+48^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+26^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6$
$n=95,160^6=115^6+106^6+104^6+101^6+99^6+96^6+95^6+94^6+93^6+92^6+90^6+89^6+88^6+86^6+85^6+83^6+82^6+81^6+80^6+79^6+78^6+77^6+76^6+75^6+74^6+73^6+72^6+71^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+63^6+62^6+61^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+49^6+48^6+47^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+6^6+5^6+4^6+3^6+2^6+1^6$ $n=100, 169^6=123^6+114^6+108^6+105^6+104^6+102^6+100^6+99^6+98^6+96^6+93^6+92^6+91^6+90^6+89^6+88^6+87^6+86^6+85^6+84^6+83^6+82^6+81^6+80^6+79^6+78^6+77^6+76^6+75^6+74^6+73^6+72^6+70^6+69^6+68^6+67^6+65^6+64^6+63^6+62^6+61^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+49^6+48^6+47^6+46^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+26^6+25^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6$
$n=101,171^6=126^6+117^6+108^6+106^6+105^6+102^6+100^6+99^6+98^6+96^6+95^6+93^6+92^6+91^6+90^6+89^6+88^6+87^6+86^6+85^6+84^6+83^6+82^6+81^6+80^6+78^6+77^6+76^6+75^6+74^6+73^6+72^6+71^6+70^6+69^6+68^6+67^6+66^6+65^6+64^6+63^6+62^6+60^6+59^6+58^6+57^6+56^6+55^6+54^6+53^6+52^6+51^6+50^6+49^6+48^6+47^6+45^6+44^6+43^6+42^6+41^6+40^6+39^6+38^6+37^6+36^6+35^6+34^6+33^6+32^6+31^6+30^6+29^6+28^6+27^6+26^6+25^6+24^6+23^6+22^6+21^6+20^6+19^6+18^6+17^6+16^6+15^6+14^6+13^6+12^6+11^6+10^6+9^6+8^6+7^6+6^6+5^6+4^6+3^6+2^6+1^6$ 同样的代码计算平方和容易很多，比如：
$n=150, 1069^2=155^2+153^2+150^2+149^2+147^2+146^2+145^2+144^2+143^2+142^2+141^2+140^2+139^2+137^2+136^2+135^2+134^2+133^2+132^2+131^2+130^2+129^2+128^2+127^2+126^2+125^2+124^2+123^2+122^2+121^2+120^2+119^2+118^2+117^2+116^2+115^2+114^2+113^2+112^2+111^2+110^2+109^2+108^2+107^2+106^2+105^2+104^2+103^2+102^2+101^2+100^2+99^2+98^2+97^2+96^2+95^2+94^2+93^2+92^2+91^2+90^2+89^2+88^2+87^2+86^2+85^2+84^2+83^2+82^2+81^2+80^2+79^2+78^2+77^2+76^2+75^2+74^2+73^2+72^2+71^2+70^2+69^2+68^2+67^2+66^2+65^2+64^2+63^2+62^2+61^2+60^2+59^2+58^2+57^2+56^2+55^2+54^2+53^2+52^2+51^2+50^2+49^2+48^2+47^2+46^2+45^2+44^2+43^2+42^2+41^2+40^2+39^2+38^2+37^2+36^2+35^2+34^2+33^2+32^2+31^2+30^2+29^2+28^2+27^2+26^2+25^2+24^2+23^2+22^2+21^2+20^2+19^2+18^2+17^2+16^2+15^2+14^2+13^2+12^2+11^2+10^2+9^2+8^2+7^2+6^2+5^2+4^2+3^2+2^2+1^2$