将正整数分解为不超过3个因子之积

northwolves · 发表于 2022-12-2 19:09:19

x=[2**200]*201
for a in range(100):
for b in range(10):
for c in range(6):
for d in range(3):
for e in range(3):
for f in range(3):
r=int((a+1)*(b+1)*(c+1)*(d+1)*(e+1)*(f+1)//2)
if r<201:
t=2**a*3**b*5**c*7**d*11**e*13**f
if x[r]>t:
x[r]=t
print(x)

复制代码

硬算的前200项，估计部分项不一定是最小的，期待mathe的优化
[1, 2, 6, 12, 24, 48, 60, 144, 120, 180, 240, 3072, 360, 900, 960, 720, 840, 5184, 1260, 36864, 1680, 2880, 3600, 12582912, 2520, 6480, 61440, 6300, 6720, 805306368, 5040, 14400, 7560, 46080, 983040, 25920, 10080, 32400, 746496, 184320, 15120, 3298534883328, 20160, 2415919104, 107520, 25200, 62914560, 21233664, 27720, 230400, 45360, 2949120, 129600, 13510798882111488, 50400, 414720, 60480, 11796480, 921600, 47775744, 55440, 9895604649984, 810000, 100800, 83160, 1658880, 322560, 176400, 6881280, 188743680, 181440, 3541774862152233910272, 110880, 1166400, 1030792151040, 226800, 14745600, 3732480, 1290240, 40532396646334464, 166320, 352800, 2073600, 14507109835375550096474112, 221760, 26542080, 65970697666560, 3240000, 967680, 928455029464035206174343168, 277200, 14929920, 440401920, 48318382080, 705600, 106168320, 332640, 8294400, 1632960, 1612800, 498960, 195689447424, 20643840, 943718400, 3870720, 907200, 67553994410557440, 356241767399424, 554400, 42501298345826806923264, 2903040, 3092376453120, 665280, 1606938044258990275541962092341162602522202993782792835301376, 82575360, 18662400, 28185722880, 6451200, 4323455642275676160, 238878720, 720720, 2822400, 29160000, 49478023249920, 112742891520, 5670000, 1108800, 132710400, 1081080, 50096498540544, 11612160, 1606938044258990275541962092341162602522202993782792835301376, 3548160, 955514880, 1106804644422573096960, 1940400, 61931520, 51840000, 1321205760, 241591910400, 1995840, 3166593487994880, 530841600, 801543976648704, 1441440, 108716359680, 70835497243044678205440, 8164800, 11289600, 1606938044258990275541962092341162602522202993782792835301376, 2494800, 1606938044258990275541962092341162602522202993782792835301376, 247726080, 103219200, 26127360, 434865438720, 14192640, 6350400, 4533471823554859405148160, 202661983231672320, 2162160, 15288238080, 3880800, 1606938044258990275541962092341162602522202993782792835301376, 115448720916480, 14515200, 15461882265600, 5976745079881894723584, 2882880, 1606938044258990275541962092341162602522202993782792835301376, 185794560, 412876800, 8493465600, 1606938044258990275541962092341162602522202993782792835301376, 84557168640, 22680000, 10644480, 12970366926827028480, 1194393600, 1606938044258990275541962092341162602522202993782792835301376, 3603600, 1606938044258990275541962092341162602522202993782792835301376, 104509440, 51881467707308113920, 3963617280, 27831388078080, 338228674560, 39690000, 7388718138654720, 7761600, 743178240, 1606938044258990275541962092341162602522202993782792835301376, 4324320, 989560464998400, 1188422437713965063903159255040, 58060800, 17962560, 24480747847196240787800064, 17740800, 4777574400, 6486480]

northwolves · 发表于 2022-12-2 19:25:41

我晕，a的上限没有改成200,只有前100项是可靠的，你可以再算一下

northwolves · 发表于 2022-12-2 19:56:56

1 2 =2^1
2 6 =2^1*3^1
3 12 =2^2*3^1
4 24 =2^3*3^1
5 48 =2^4*3^1
6 60 =2^2*3^1*5^1
7 144 =2^4*3^2
8 120 =2^3*3^1*5^1
9 180 =2^2*3^2*5^1
10 240 =2^4*3^1*5^1
11 3072 =2^10*3^1
12 360 =2^3*3^2*5^1
13 900 =2^2*3^2*5^2
14 960 =2^6*3^1*5^1
15 720 =2^4*3^2*5^1
16 840 =2^3*3^1*5^1*7^1
17 5184 =2^6*3^4
18 1260 =2^2*3^2*5^1*7^1
19 36864 =2^12*3^2
20 1680 =2^4*3^1*5^1*7^1
21 2880 =2^6*3^2*5^1
22 3600 =2^4*3^2*5^2
23 12582912 =2^22*3^1
24 2520 =2^3*3^2*5^1*7^1
25 6480 =2^4*3^4*5^1
26 61440 =2^12*3^1*5^1
27 6300 =2^2*3^2*5^2*7^1
28 6720 =2^6*3^1*5^1*7^1
29 805306368 =2^28*3^1
30 5040 =2^4*3^2*5^1*7^1
31 14400 =2^6*3^2*5^2
32 7560 =2^3*3^3*5^1*7^1
33 46080 =2^10*3^2*5^1
34 983040 =2^16*3^1*5^1
35 25920 =2^6*3^4*5^1
36 10080 =2^5*3^2*5^1*7^1
37 32400 =2^4*3^4*5^2
38 746496 =2^10*3^6
39 184320 =2^12*3^2*5^1
40 15120 =2^4*3^3*5^1*7^1
41 3298534883328 =2^40*3^1
42 20160 =2^6*3^2*5^1*7^1
43 2415919104 =2^28*3^2
44 107520 =2^10*3^1*5^1*7^1
45 25200 =2^4*3^2*5^2*7^1
46 62914560 =2^22*3^1*5^1
47 21233664 =2^18*3^4
48 27720 =2^3*3^2*5^1*7^1*11^1
49 230400 =2^10*3^2*5^2
50 45360 =2^4*3^4*5^1*7^1
51 2949120 =2^16*3^2*5^1
52 129600 =2^6*3^4*5^2
53 13510798882111488 =2^52*3^1
54 50400 =2^5*3^2*5^2*7^1
55 414720 =2^10*3^4*5^1
56 60480 =2^6*3^3*5^1*7^1
57 11796480 =2^18*3^2*5^1
58 921600 =2^12*3^2*5^2
59 47775744 =2^16*3^6
60 55440 =2^4*3^2*5^1*7^1*11^1
61 9895604649984 =2^40*3^2
62 810000 =2^4*3^4*5^4
63 100800 =2^6*3^2*5^2*7^1
64 83160 =2^3*3^3*5^1*7^1*11^1
65 1658880 =2^12*3^4*5^1
66 322560 =2^10*3^2*5^1*7^1
67 176400 =2^4*3^2*5^2*7^2
68 6881280 =2^16*3^1*5^1*7^1
69 188743680 =2^22*3^2*5^1
70 181440 =2^6*3^4*5^1*7^1
71 241864704 =2^12*3^10
72 110880 =2^5*3^2*5^1*7^1*11^1
73 1166400 =2^6*3^6*5^2
74 1030792151040 =2^36*3^1*5^1
75 226800 =2^4*3^4*5^2*7^1
76 14745600 =2^16*3^2*5^2
77 3732480 =2^10*3^6*5^1
78 1290240 =2^12*3^2*5^1*7^1
79 40532396646334464 =2^52*3^2
80 166320 =2^4*3^3*5^1*7^1*11^1
81 352800 =2^5*3^2*5^2*7^2
82 2073600 =2^10*3^4*5^2
83 14507109835375550096474112 =2^82*3^1
84 221760 =2^6*3^2*5^1*7^1*11^1
85 26542080 =2^16*3^4*5^1
86 65970697666560 =2^42*3^1*5^1
87 3240000 =2^6*3^4*5^4
88 967680 =2^10*3^3*5^1*7^1
89 928455029464035206174343168 =2^88*3^1
90 277200 =2^4*3^2*5^2*7^1*11^1
91 14929920 =2^12*3^6*5^1
92 440401920 =2^22*3^1*5^1*7^1
93 3869835264 =2^16*3^10
94 705600 =2^6*3^2*5^2*7^2
95 106168320 =2^18*3^4*5^1
96 332640 =2^5*3^3*5^1*7^1*11^1
97 8294400 =2^12*3^4*5^2
98 1632960 =2^6*3^6*5^1*7^1
99 1612800 =2^10*3^2*5^2*7^1
100 498960 =2^4*3^4*5^1*7^1*11^1
101 195689447424 =2^28*3^6
102 20643840 =2^16*3^2*5^1*7^1
103 943718400 =2^22*3^2*5^2
104 3870720 =2^12*3^3*5^1*7^1
105 907200 =2^6*3^4*5^2*7^1
106 67553994410557440 =2^52*3^1*5^1
107 356241767399424 =2^42*3^4
108 554400 =2^5*3^2*5^2*7^1*11^1
109 42501298345826806923264 =2^72*3^2
110 2903040 =2^10*3^4*5^1*7^1
111 3092376453120 =2^36*3^2*5^1
112 665280 =2^6*3^3*5^1*7^1*11^1
113 15576890575604482885591488987660288 =2^112*3^1
114 82575360 =2^18*3^2*5^1*7^1
115 18662400 =2^10*3^6*5^2
116 28185722880 =2^28*3^1*5^1*7^1
117 6451200 =2^12*3^2*5^2*7^1
118 4323455642275676160 =2^58*3^1*5^1
119 238878720 =2^16*3^6*5^1
120 720720 =2^4*3^2*5^1*7^1*11^1*13^1
121 2822400 =2^8*3^2*5^2*7^2
122 29160000 =2^6*3^6*5^4
123 139314069504 =2^18*3^12
124 112742891520 =2^30*3^1*5^1*7^1
125 5670000 =2^4*3^4*5^4*7^1
126 1108800 =2^6*3^2*5^2*7^1*11^1
127 132710400 =2^16*3^4*5^2
128 1081080 =2^3*3^3*5^1*7^1*11^1*13^1
129 50096498540544 =2^36*3^6
130 11612160 =2^12*3^4*5^1*7^1
131 4083388403051261561560495289181218537472 =2^130*3^1
132 3548160 =2^10*3^2*5^1*7^1*11^1
133 955514880 =2^18*3^6*5^1
134 1106804644422573096960 =2^66*3^1*5^1
135 1940400 =2^4*3^2*5^2*7^2*11^1
136 61931520 =2^16*3^3*5^1*7^1
137 51840000 =2^10*3^4*5^4
138 1321205760 =2^22*3^2*5^1*7^1
139 241591910400 =2^30*3^2*5^2
140 1995840 =2^6*3^4*5^1*7^1*11^1
141 3166593487994880 =2^46*3^2*5^1
142 530841600 =2^18*3^4*5^2
143 1209323520 =2^12*3^10*5^1
144 1441440 =2^5*3^2*5^1*7^1*11^1*13^1
145 108716359680 =2^28*3^4*5^1
146 70835497243044678205440 =2^72*3^1*5^1
147 8164800 =2^6*3^6*5^2*7^1
148 11289600 =2^10*3^2*5^2*7^2
149 2229025112064 =2^22*3^12
150 2494800 =2^4*3^4*5^2*7^1*11^1
151 11408855402054064613470328848384 =2^100*3^2
152 247726080 =2^18*3^3*5^1*7^1
153 103219200 =2^16*3^2*5^2*7^1
154 26127360 =2^10*3^6*5^1*7^1
155 434865438720 =2^30*3^4*5^1
156 14192640 =2^12*3^2*5^1*7^1*11^1
157 6350400 =2^6*3^4*5^2*7^2
158 4533471823554859405148160 =2^78*3^1*5^1
159 15850845241344 =2^28*3^10
160 2162160 =2^4*3^3*5^1*7^1*11^1*13^1
161 15288238080 =2^22*3^6*5^1
162 3880800 =2^5*3^2*5^2*7^2*11^1
163 2920666982925840541048404185186304 =2^108*3^2
164 115448720916480 =2^40*3^1*5^1*7^1
165 14515200 =2^10*3^4*5^2*7^1
166 15461882265600 =2^36*3^2*5^2
167 5976745079881894723584 =2^66*3^4
168 2882880 =2^6*3^2*5^1*7^1*11^1*13^1
169 10883911680 =2^12*3^12*5^1
170 185794560 =2^16*3^4*5^1*7^1
171 412876800 =2^18*3^2*5^2*7^1
172 8493465600 =2^22*3^4*5^2
173 17958932119522135058886879224417685745532099088089088 =2^172*3^1
174 84557168640 =2^28*3^2*5^1*7^1
175 22680000 =2^6*3^4*5^4*7^1
176 10644480 =2^10*3^3*5^1*7^1*11^1
177 12970366926827028480 =2^58*3^2*5^1
178 1194393600 =2^16*3^6*5^2
179 1149371655649416643768760270362731887714054341637701632 =2^178*3^1
180 3603600 =2^4*3^2*5^2*7^1*11^1*13^1
181 1511654400 =2^10*3^10*5^2
182 104509440 =2^12*3^6*5^1*7^1
183 51881467707308113920 =2^60*3^2*5^1
184 3963617280 =2^22*3^3*5^1*7^1
185 27831388078080 =2^36*3^4*5^1
186 338228674560 =2^30*3^2*5^1*7^1
187 39690000 =2^4*3^4*5^4*7^2
188 142657607172096 =2^28*3^12
189 7761600 =2^6*3^2*5^2*7^2*11^1
190 743178240 =2^18*3^4*5^1*7^1
191 4707826301540010572876842067405749812076766583348025884672 =2^190*3^1
192 4324320 =2^5*3^3*5^1*7^1*11^1*13^1
193 989560464998400 =2^42*3^2*5^2
194 1188422437713965063903159255040 =2^96*3^1*5^1
195 58060800 =2^12*3^4*5^2*7^1
196 17962560 =2^6*3^6*5^1*7^1*11^1
197 24480747847196240787800064 =2^78*3^4
198 17740800 =2^10*3^2*5^2*7^1*11^1
199 4777574400 =2^18*3^6*5^2
200 6486480 =2^4*3^4*5^1*7^1*11^1*13^1

northwolves · 发表于 2022-12-2 20:30:58

本帖最后由 northwolves 于 2022-12-2 20:34 编辑

$对于n=p_1^{t_1}p_2^{t_2}...p_i^{t_i}...p_r^{t_r}$
$易知 \tao(n)=(1+t_1)(1+t_2)...(1+t_i)...(1+t_r)//2$
$a(n)就是找到这样的最小的n对应的t_i,$
$似乎对于大多数的素数p,我们有 a(p)=2^{p-1}*3$

northwolves · 发表于 2022-12-3 12:34:22

$对于素数p,若2p+1也是素数，则 a(p)=2^{p-1}*3$

王守恩 · 发表于 2022-12-3 13:14:17

将一个正整数恰好分解为 4 个因子之积, 可以有 n 种分解方式，我们取其中最小的那个数。

a(01)=01: 1×1×1×01,
a(02)=04: 1×1×1×04, 1×1×2×02,
a(03)=08: 1×1×1×08, 1×1×2×04, 1×2×2×02,
a(04)=12: 1×1×1×12, 1×1×2×06, 1×1×3×04, 1×2×2×3,
a(05)=16: 1×1×1×16, 1×1×2×08, 1×1×4×04, 1×2×2×4, 2×2×2×2,
a(06)=32: 1×1×1×32, 1×1×2×16, 1×1×4×08, 1×2×2×8, 1×2×4×4, 2×2×2×4,
a(07)=24: 1×1×1×24, 1×1×2×12, 1×1×3×08, 1×1×4×6, 1×2×2×6, 1×2×3×4, 2×2×2×3,
a(08)=00:
a(09)=36: 1×1×1×36, 1×1×2×18, 1×1×3×12, 1×1×4×9, 1×1×6×6, 1×2×2×9, 1×2×3×6, 1×3×3×4, 2×2×3×3,

得到一个在 OEIS 找不到的整数序列

1, 4, 8, 12, 16, 32, 24, 0, 36, 0, 48, 0, 0, 0, 72, 96, 0, 512, 0, 120, 0, 0, 192, 0, 144, 0, 216, 0, 0, 0, 84, 0, 336, 0, 420, 288, ......

通项公式出不了。观察发现: 如果项数是下面序列中的数，则肯定有解。譬如:项数=1, 2, 3, 4, 5, 6, 9, 11, 15, 18, 23, 27, ......

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270,
297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, ......

$a(n)=\lceil\frac{2 n^3 + 30 n^2 + 135 n + 31 + \cos(n\pi)(9 n + 45)}{288}\rceil$

王守恩 · 发表于 2022-12-3 13:52:06

一个正整数的倒数,化成循环小数,循环节长度是一定的，我们取其中最小的那个数。

a(1)=3, a(2)=11, a(3)=37, a(4)=101, a(5)=41, a(6)=7,......, a(100)=60101,......

得到一串数：

3, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19,
1111111111111111111, 3541, 43, 23, 11111111111111111111111,  99990001, 21401, 859,
757, 29, 3191, 211, 2791, 353, 67, 103,  71,  999999000001,  2028119,  909090909090909091,
900900900900990990990991,1676321,83,127,173,89,238681,47,35121409,9999999900000001,
505885997,  251, 613, 521, 107,  70541929, 1321, 7841, 21319,  59, 2559647034361,  61, ......

A007138

18世纪，给出了前16项
2010年6月1日给出前276行项
2017年5月1日，给出了前322项
2022年4月26日，给出了前352项
最新：文件被截断为前364项，因为a(365)被发现是错误的

各位网友:我们可以往前推一推吗？

northwolves · 发表于 2022-12-3 16:11:21

a(365)=499024823519?
a(365)=3298432370089?

northwolves · 发表于 2022-12-3 17:31:51

northwolves 发表于 2022-12-3 16:11
a(365)=499024823519?
a(365)=3298432370089?

499024823519=41*12171337159

mathe · 发表于 2022-12-3 18:59:22

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[讨论] 将正整数分解为不超过3个因子之积

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