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数学研发论坛»论坛 › 【数学研究】 › 小题大作 › 数字串的通项公式
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楼主: 王守恩

[原创] 数字串的通项公式

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王守恩
 楼主| 发表于 2025-12-6 10:52:10 | 显示全部楼层
{1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60}
Table[EulerPhi[n], {n, 61}]

{1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60}
Table[Sum[Boole[CoprimeQ[k, n]], {k, n}], {n, 61}]——两者是一样的。
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王守恩
 楼主| 发表于 2025-12-9 11:45:20 | 显示全部楼层
northwolves 发表于 2025-9-27 09:29

a(1)=5,5^2=3^2+4^2,

a(2)=25,25^2=7^2+24^2=15^2+20^2,

a(3)=65,65^2=16^2+63^2=25^2+60^2=33^2+56^2,

a(4)=65,65^2=16^2+63^2=25^2+60^2=33^2+56^2=39^2+52^2,

a(5)=325,325^2=36^2+323^2=80^2+315^2=91^2+312^2=125^2+300^2=165^2+280^2,

a(6)=325,325^2=36^2+323^2=80^2+315^2=91^2+312^2=125^2+300^2=165^2+280^2=195^2+260^2,

a(7)=325,325^2=36^2+323^2=80^2+315^2=91^2+312^2=125^2+300^2=165^2+280^2=195^2+260^2=204^2+253^2,

a(8)=1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2,

a(9)=1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2=561^2+952^2,

a(10)=1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2=561^2+952^2=576^2+943^2,

a(11)=1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2=561^2+952^2=576^2+943^2=663^2+884^2,

a(12)==1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2=561^2+952^2=576^2+943^2=663^2+884^2=700^2+855^2,

a(13)==1105,1105^2=47^2+1104^2=105^2+1100^2=169^2+1092^2=264^2+1073^2=272^2+1071^2=425^2+1020^2=468^2+1001^2=520^2+975^2=561^2+952^2=576^2+943^2=663^2+884^2=700^2+855^2=744^2+817^2,
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northwolves
发表于 2025-12-9 19:47:37 | 显示全部楼层
王守恩 发表于 2025-12-9 11:45
a(1)=5,5^2=3^2+4^2,

a(2)=25,25^2=7^2+24^2=15^2+20^2,

A088959
Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of two positive squares in more ways than for any smaller number.

1, 5, 25, 65, 325, 1105, 5525, 27625, 32045, 160225, 801125, 1185665, 5928325, 29641625, 48612265, 243061325, 1215306625, 2576450045, 12882250225, 64411251125, 157163452745, 785817263725, 3929086318625, 10215624428425, 11472932050385, 51078122142125
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northwolves
发表于 2025-12-9 19:59:37 | 显示全部楼层
  1. s={5,25,65,325,1105,5525,27625,32045};t=Length@Values@Solve[{x^2+y^2==#^2,0<x<y},Integers]&/@s;a=Flatten@MapThread[ConstantArray[#1,#2]&,{s,Prepend[Differences[t],t[[1]]]}];Do[Print[{k,a[[k]],Take[Values@Solve[{x^2+y^2==a[[k]]^2,0<x<y},Integers],k]}],{k,Length@a}]
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{1,5,{{3,4}}}
{2,25,{{7,24},{15,20}}}
{3,65,{{16,63},{25,60},{33,56}}}
{4,65,{{16,63},{25,60},{33,56},{39,52}}}
{5,325,{{36,323},{80,315},{91,312},{125,300},{165,280}}}
{6,325,{{36,323},{80,315},{91,312},{125,300},{165,280},{195,260}}}
{7,325,{{36,323},{80,315},{91,312},{125,300},{165,280},{195,260},{204,253}}}
{8,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975}}}
{9,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975},{561,952}}}
{10,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975},{561,952},{576,943}}}
{11,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975},{561,952},{576,943},{663,884}}}
{12,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975},{561,952},{576,943},{663,884},{700,855}}}
{13,1105,{{47,1104},{105,1100},{169,1092},{264,1073},{272,1071},{425,1020},{468,1001},{520,975},{561,952},{576,943},{663,884},{700,855},{744,817}}}
{14,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875}}}
{15,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760}}}
{16,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715}}}
{17,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557}}}
{18,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557},{3315,4420}}}
{19,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557},{3315,4420},{3468,4301}}}
{20,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557},{3315,4420},{3468,4301},{3500,4275}}}
{21,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557},{3315,4420},{3468,4301},{3500,4275},{3720,4085}}}
{22,5525,{{235,5520},{525,5500},{612,5491},{845,5460},{1036,5427},{1131,5408},{1320,5365},{1360,5355},{1547,5304},{2044,5133},{2125,5100},{2163,5084},{2340,5005},{2600,4875},{2805,4760},{2880,4715},{3124,4557},{3315,4420},{3468,4301},{3500,4275},{3720,4085},{3861,3952}}}
{23,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575}}}
{24,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785}}}
{25,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100}}}
{26,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505}}}
{27,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505},{17500,21375}}}
{28,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505},{17500,21375},{18239,20748}}}
{29,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505},{17500,21375},{18239,20748},{18600,20425}}}
{30,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505},{17500,21375},{18239,20748},{18600,20425},{18921,20128}}}
{31,27625,{{969,27608},{1175,27600},{2625,27500},{3060,27455},{3588,27391},{4225,27300},{5180,27135},{5655,27040},{6600,26825},{6800,26775},{7223,26664},{7735,26520},{8856,26167},{9724,25857},{10220,25665},{10625,25500},{10815,25420},{11700,25025},{12137,24816},{13000,24375},{13847,23904},{14025,23800},{14400,23575},{15620,22785},{16575,22100},{17340,21505},{17500,21375},{18239,20748},{18600,20425},{18921,20128},{19305,19760}}}
{32,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636}}}
{33,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389}}}
{34,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200}}}
{35,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795}}}
{36,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795},{21000,24205}}}
{37,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795},{21000,24205},{21093,24124}}}
{38,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795},{21000,24205},{21093,24124},{21576,23693}}}
{39,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795},{21000,24205},{21093,24124},{21576,23693},{22100,23205}}}
{40,32045,{{716,32037},{1363,32016},{2277,31964},{2400,31955},{3045,31900},{3757,31824},{3955,31800},{4901,31668},{5304,31603},{6764,31323},{7259,31212},{7656,31117},{7888,31059},{8283,30956},{8580,30875},{8772,30821},{10075,30420},{10192,30381},{11475,29920},{11661,29848},{12325,29580},{12920,29325},{13572,29029},{15080,28275},{15708,27931},{15916,27813},{16269,27608},{16704,27347},{17051,27132},{17253,27004},{18291,26312},{19227,25636},{19552,25389},{19795,25200},{20300,24795},{21000,24205},{21093,24124},{21576,23693},{22100,23205},{22244,23067}}}
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 2025-12-10 08:46:25 | 显示全部楼层
northwolves 发表于 2025-12-9 19:59
{1,5,{{3,4}}}
{2,25,{{7,24},{15,20}}}
{3,65,{{16,63},{25,60},{33,56}}}

A046080——{0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0,

可以这样——Table[Length[n /. Solve[{n^2 == a^2 + b^2, n == k, a > b > 0}, {n, a, b}, Integers]], {k, 250}]——我就是不知道怎么把第1个数拉出来变成

5, 25, 125, 65, 3125, 15625, 325, 390625, 1953125, 1625, 48828125, 4225, 1105, 6103515625, 30517578125, 40625, 21125, 3814697265625, 203125, 95367431640625,——A006339——Least hypotenuse of n distinct Pythagorean triangles.

又。Union[n /. Solve[{a^2 + b^2 == n^2, 100 > n > 0, a > b > 0}, {n, a, b}, Integers]]
{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97}
Select[Range[99], SquaresR[2, #^2] > 4 &]
{5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 51, 52, 53, 55, 58, 60, 61, 65, 68, 70, 73, 74, 75, 78, 80, 82, 85, 87, 89, 90, 91, 95, 97}

Select[Range[100], SquaresR[2, #^3] > 4 &]——代码 A。
{5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 90, 97, 100}
Select[Range@100, Length[PowersRepresentations[#, 2, 2] /. {{0, _} -> Nothing, {a_, b_} /; a == b -> Nothing}] > 0 &]——代码 B。
{5, 10, 13, 17, 20, 25, 26, 29, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 73, 74, 80, 82, 85, 89, 90, 97, 100}——A004431——Numbers that are the sum of 2 distinct nonzero squares.
问:  代码 A 与代码 B 有区别吗？  代码 A 可以替代代码 B 吗？
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王守恩
 楼主| 发表于 2025-12-10 09:02:20 | 显示全部楼层
northwolves 发表于 2025-12-9 19:47
A088959
Lowest numbers which are d-Pythagorean decomposable, i.e., square is expressible as sum of ...

\(n^2=a_{1}^2+b_{1}^2=a_{2}^2+b_{2}^2=a_{3}^2+b_{3}^2=\cdots\cdots=a_{2026}^2+b_{2026}^2\)

1,  n 可以写成 2026 种两平方和(正整数无序对),  n 最小=____。

2,  n 恰好写成 2026 种两平方和(正整数无序对),  n 最小=____。
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northwolves
发表于 2025-12-10 14:48:51 | 显示全部楼层
n 可以写成 2026 种两平方和(正整数无序对),  n 最小=64411251125.
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northwolves
发表于 2025-12-10 15:03:40 | 显示全部楼层
n 恰好写成 2026 种两平方和(正整数无序对),  n 最小=471410655185593118725162402955948526273033394318190403282642364501953125
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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northwolves
发表于 2025-12-10 15:38:54 | 显示全部楼层
northwolves 发表于 2025-12-10 15:03
n 恰好写成 2026 种两平方和(正整数无序对),  n 最小=47141065518559311872516240295594852627303339431819 ...
  1. f[n_]:=Module[{p={5},g},g[1,_]=1;
  2. g[m_,k_]:=g[m,k]=Min[(While[Length[p]<k,AppendTo[p,NestWhile[NextPrime,NextPrime[Last[p]],Mod[#,4]!=1&]]];
  3. p[[k]])^((#-1)/2)*g[m/#,k+1]&/@Rest[Divisors[m]]];
  4. g[2 n+1,1]];f[2026]
复制代码
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王守恩
 楼主| 发表于 2025-12-10 16:40:10 | 显示全部楼层
northwolves 发表于 2025-12-10 15:03
n 恰好写成 2026 种两平方和(正整数无序对),  n 最小=47141065518559311872516240295594852627303339431819 ...

Table[Length[n /. Solve[{n^2 == a^2 + b^2, k == n, a > b > 0}, {n, a, b}, Integers]],
{k, 471410655185593118725162402955948526273033394318190403282642364501953125, 471410655185593118725162402955948526273033394318190403282642364501953125}]
{2026}

补充内容 (2025-12-14 12:36):
Table[{Length@
   Select[PowersRepresentations[k^2, 2, 2], #[[1]] > 0 &]}, {k,——这个速度还慢一些？
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毋私小惠而伤大体　　毋借公论以快私情
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