[原创]方程2x+3y+5z+7u=N的正整数解的组数系列公式

白新岭

[watermark]MOD(N,210)→→at^3→→bt^2→→ct→→d
1→→44100→→-4725→→147→→0
2→→44100→→-4095→→105→→0
3→→44100→→-3465→→69→→0
4→→44100→→-2835→→39→→0
5→→44100→→-2205→→15→→0
6→→44100→→-1575→→-3→→0
7→→44100→→-945→→-15→→0
8→→44100→→-315→→-21→→0
9→→44100→→315→→-21→→0
10→→44100→→945→→-15→→0
11→→44100→→1575→→-3→→0
12→→44100→→2205→→15→→0
13→→44100→→2835→→39→→0
14→→44100→→3465→→69→→0
15→→44100→→4095→→105→→0
16→→44100→→4725→→147→→0
17→→44100→→5355→→195→→6
18→→44100→→5985→→249→→0
19→→44100→→6615→→309→→6
20→→44100→→7245→→375→→6
21→→44100→→7875→→447→→6
22→→44100→→8505→→525→→12
23→→44100→→9135→→609→→12
24→→44100→→9765→→699→→18
25→→44100→→10395→→795→→18
26→→44100→→11025→→897→→24
27→→44100→→11655→→1005→→30
28→→44100→→12285→→1119→→30
29→→44100→→12915→→1239→→42
30→→44100→→13545→→1365→→42
31→→44100→→14175→→1497→→54
32→→44100→→14805→→1635→→60
33→→44100→→15435→→1779→→66
34→→44100→→16065→→1929→→78
35→→44100→→16695→→2085→→84
36→→44100→→17325→→2247→→96
37→→44100→→17955→→2415→→108
38→→44100→→18585→→2589→→120
39→→44100→→19215→→2769→→132
40→→44100→→19845→→2955→→144
41→→44100→→20475→→3147→→162
42→→44100→→21105→→3345→→174
43→→44100→→21735→→3549→→192
44→→44100→→22365→→3759→→210
45→→44100→→22995→→3975→→228
46→→44100→→23625→→4197→→246
47→→44100→→24255→→4425→→270
48→→44100→→24885→→4659→→288
49→→44100→→25515→→4899→→312
50→→44100→→26145→→5145→→336
51→→44100→→26775→→5397→→360
52→→44100→→27405→→5655→→390
53→→44100→→28035→→5919→→414
54→→44100→→28665→→6189→→444
55→→44100→→29295→→6465→→474
56→→44100→→29925→→6747→→504
57→→44100→→30555→→7035→→540
58→→44100→→31185→→7329→→570
59→→44100→→31815→→7629→→612
60→→44100→→32445→→7935→→642
61→→44100→→33075→→8247→→684
62→→44100→→33705→→8565→→726
63→→44100→→34335→→8889→→762
64→→44100→→34965→→9219→→810
65→→44100→→35595→→9555→→852
66→→44100→→36225→→9897→→900
67→→44100→→36855→→10245→→948
68→→44100→→37485→→10599→→996
69→→44100→→38115→→10959→→1050
70→→44100→→38745→→11325→→1098
71→→44100→→39375→→11697→→1158
72→→44100→→40005→→12075→→1212
73→→44100→→40635→→12459→→1272
74→→44100→→41265→→12849→→1332
75→→44100→→41895→→13245→→1392
76→→44100→→42525→→13647→→1458
77→→44100→→43155→→14055→→1524
78→→44100→→43785→→14469→→1590
79→→44100→→44415→→14889→→1662
80→→44100→→45045→→15315→→1734
81→→44100→→45675→→15747→→1806
82→→44100→→46305→→16185→→1884
83→→44100→→46935→→16629→→1962
84→→44100→→47565→→17079→→2040
85→→44100→→48195→→17535→→2124
86→→44100→→48825→→17997→→2208
87→→44100→→49455→→18465→→2298
88→→44100→→50085→→18939→→2382
89→→44100→→50715→→19419→→2478
90→→44100→→51345→→19905→→2568
91→→44100→→51975→→20397→→2664
92→→44100→→52605→→20895→→2766
93→→44100→→53235→→21399→→2862
94→→44100→→53865→→21909→→2970
95→→44100→→54495→→22425→→3072
96→→44100→→55125→→22947→→3180
97→→44100→→55755→→23475→→3294
98→→44100→→56385→→24009→→3402
99→→44100→→57015→→24549→→3522
100→→44100→→57645→→25095→→3636
101→→44100→→58275→→25647→→3762
102→→44100→→58905→→26205→→3882
103→→44100→→59535→→26769→→4008
104→→44100→→60165→→27339→→4140
105→→44100→→60795→→27915→→4266
106→→44100→→61425→→28497→→4404
107→→44100→→62055→→29085→→4542
108→→44100→→62685→→29679→→4680
109→→44100→→63315→→30279→→4824
110→→44100→→63945→→30885→→4968
111→→44100→→64575→→31497→→5118
112→→44100→→65205→→32115→→5268
113→→44100→→65835→→32739→→5424
114→→44100→→66465→→33369→→5580
115→→44100→→67095→→34005→→5742
116→→44100→→67725→→34647→→5904
117→→44100→→68355→→35295→→6072
118→→44100→→68985→→35949→→6240
119→→44100→→69615→→36609→→6414
120→→44100→→70245→→37275→→6588
121→→44100→→70875→→37947→→6768
122→→44100→→71505→→38625→→6954
123→→44100→→72135→→39309→→7134
124→→44100→→72765→→39999→→7326
125→→44100→→73395→→40695→→7518
126→→44100→→74025→→41397→→7710
127→→44100→→74655→→42105→→7914
128→→44100→→75285→→42819→→8112
129→→44100→→75915→→43539→→8322
130→→44100→→76545→→44265→→8526
131→→44100→→77175→→44997→→8742
132→→44100→→77805→→45735→→8958
133→→44100→→78435→→46479→→9174
134→→44100→→79065→→47229→→9402
135→→44100→→79695→→47985→→9624
136→→44100→→80325→→48747→→9858
137→→44100→→80955→→49515→→10092
138→→44100→→81585→→50289→→10326
139→→44100→→82215→→51069→→10572
140→→44100→→82845→→51855→→10812
141→→44100→→83475→→52647→→11064
142→→44100→→84105→→53445→→11316
143→→44100→→84735→→54249→→11574
144→→44100→→85365→→55059→→11832
145→→44100→→85995→→55875→→12096
146→→44100→→86625→→56697→→12366
147→→44100→→87255→→57525→→12636
148→→44100→→87885→→58359→→12912
149→→44100→→88515→→59199→→13194
150→→44100→→89145→→60045→→13476
151→→44100→→89775→→60897→→13764
152→→44100→→90405→→61755→→14058
153→→44100→→91035→→62619→→14352
154→→44100→→91665→→63489→→14652
155→→44100→→92295→→64365→→14958
156→→44100→→92925→→65247→→15264
157→→44100→→93555→→66135→→15582
158→→44100→→94185→→67029→→15894
159→→44100→→94815→→67929→→16218
160→→44100→→95445→→68835→→16542
161→→44100→→96075→→69747→→16872
162→→44100→→96705→→70665→→17208
163→→44100→→97335→→71589→→17544
164→→44100→→97965→→72519→→17892
165→→44100→→98595→→73455→→18234
166→→44100→→99225→→74397→→18588
167→→44100→→99855→→75345→→18948
168→→44100→→100485→→76299→→19302
169→→44100→→101115→→77259→→19674
170→→44100→→101745→→78225→→20040
171→→44100→→102375→→79197→→20418
172→→44100→→103005→→80175→→20796
173→→44100→→103635→→81159→→21180
174→→44100→→104265→→82149→→21570
175→→44100→→104895→→83145→→21960
176→→44100→→105525→→84147→→22362
177→→44100→→106155→→85155→→22764
178→→44100→→106785→→86169→→23172
179→→44100→→107415→→87189→→23586
180→→44100→→108045→→88215→→24000
181→→44100→→108675→→89247→→24426
182→→44100→→109305→→90285→→24852
183→→44100→→109935→→91329→→25284
184→→44100→→110565→→92379→→25722
185→→44100→→111195→→93435→→26166
186→→44100→→111825→→94497→→26610
187→→44100→→112455→→95565→→27066
188→→44100→→113085→→96639→→27522
189→→44100→→113715→→97719→→27984
190→→44100→→114345→→98805→→28452
191→→44100→→114975→→99897→→28926
192→→44100→→115605→→100995→→29406
193→→44100→→116235→→102099→→29886
194→→44100→→116865→→103209→→30378
195→→44100→→117495→→104325→→30870
196→→44100→→118125→→105447→→31368
197→→44100→→118755→→106575→→31878
198→→44100→→119385→→107709→→32382
199→→44100→→120015→→108849→→32904
200→→44100→→120645→→109995→→33420
201→→44100→→121275→→111147→→33948
202→→44100→→121905→→112305→→34482
203→→44100→→122535→→113469→→35016
204→→44100→→123165→→114639→→35562
205→→44100→→123795→→115815→→36108
206→→44100→→124425→→116997→→36666
207→→44100→→125055→→118185→→37224
208→→44100→→125685→→119379→→37788
209→→44100→→126315→→120579→→38364
0→→44100→→126945→→121785→→38934
[/watermark]

白新岭

主贴中余数（对模210的余数）所对应行为方程的正整数解的组数公式中的系数，标准形式为：(at^3+bt^2+ct+d)/6,t=INT((N-1)/210).例如N=2010,MOD(2010,210)=120,找到余数120对应行的系数，t=INT((2010-1)/210)=9,带入公式120→→（44100*t^3+70245*t^2+37275*t+6588）/6=6363468
所以，2x+3y+5z+7u=2010有6363468组正整数解。