那一个数的多少次方后包含有1234567890这10个数字

香川群子 · 发表于 2011-9-13 22:46

计算搜寻了一把，a^b这样形式的，计算结果按0-9数字的有很多。

a取3-99，b取6-99，共97*94=9118种结果，
分析其中符合条件的共有=7479个，占82%

其中位数最短的是13位数=56^7=1727094849536
其中位数最长的当然是最后的是198位
=99^99=369729637649726772657187905628805440595668764281741102430259972423552570455277523421410650010128232727940978889548326540119429996769494359451621570193644014418071060667659301384999779999159200499899

属于a^a形式(数字自身幂)的一共有73个，它们是：
=19^19 1978419655660313589123979
=23^23 20880467999847912034355032910567
=24^24 1333735776850284124449081472843776
=25^25 88817841970012523233890533447265625
=26^26 6156119580207157310796674288400203776
=29^29 2567686153161211134561828214731016126483469
=31^31 17069174130723235958610643029059314756044734431
=32^32 1461501637330902918203684832716283019655932542976
=33^33 129110040087761027839616029934664535539337183380513
=35^35 1102507499354148695951786433413508348166942596435546875
=36^36 106387358923716524807713475752456393740167855629859291136
=37^37 10555134955777783414078330085995832946127396083370199442517
=38^38 1075911801979993982060429252856123779115487368830416064610304
=39^39 112595147462071192539789448988889059930192105219196517009951959
=41^41 1330877630632711998713399240963346255985889330161650994325137953641

以下结果略：
=42^42
=43^43
=44^44
=45^45
=46^46
=47^47
=48^48
=49^49
=50^50
=51^51
=52^52
=53^53
=54^54
=55^55
=56^56
=57^57
=58^58
=59^59
=60^60
=61^61
=62^62
=63^63
=64^64
=65^65
=66^66
=67^67
=68^68
=69^69
=70^70
=71^71
=72^72
=73^73
=74^74
=75^75
=76^76
=77^77
=78^78
=79^79
=80^80
=81^81
=82^82
=83^83
=84^84
=85^85
=86^86
=87^87
=88^88
=89^89
=90^90
=91^91
=92^92
=93^93
=94^94
=95^95
=96^96
=97^97
=98^98
=99^99

香川群子 · 发表于 2011-9-13 22:49

结论，从41开始，任何整数n的n次方结果中，必定含有0-9这10个数字。

wangg913 · 发表于 2011-9-13 22:52

你把简单问题搞复杂了，这没什么意义。

香川群子 · 发表于 2011-9-13 23:00

结果位数<=20的44个结果：

=56^7 1727094849536
=5^19 19073486328125
=15^12 129746337890625
=49^9 1628413597910449
=7^18 1628413597910449
=25^11 2384185791015625
=5^22 2384185791015625
=36^10 3656158440062976
=6^20 3656158440062976
=28^11 8293509467471872
=6^21 21936950640377856
=31^11 25408476896404831
=68^9 31087100296429568
=82^9 167619550409708032
=17^14 168377826559400929
=83^9 186940255267540403
=84^9 208215748530929664
=54^10 210832519264920576
=29^12 353814783205469041
=41^11 550329031716248441
=99^9 913517247483640899
=65^10 1346274334462890625
=25^13 1490116119384765625
=5^26 1490116119384765625
=67^10 1822837804551761449
=46^11 1951354384207722496
=68^10 2113922820157210624
=17^15 2862423051509815793
=27^13 4052555153018976267
=3^39 4052555153018976267
=15^16 6568408355712890625
=5^27 7450580596923828125
=52^11 7516865509350965248
=23^14 11592836324538749809
=82^10 13744803133596058624
=83^10 15516041187205853449
=85^10 19687440434072265625
=58^11 24986644000165537792
=59^11 30155888444737842659
=25^14 37252902984619140625
=5^28 37252902984619140625
=96^10 66483263599150104576
=65^11 87507831740087890625
=46^12 89762301673555234816

bluexuemei · 发表于 2011-9-13 23:22

香川群子发表于 2011-9-13 23:00
结果位数

学习群子的研究精神,佩服!

香川群子 · 发表于 2011-9-13 23:33

最后，再来一些除法的：
a=1-9, b=3-99,计算结果符合条件的有：

=1/17 0.0588235294117647
=1/19 0.0526315789473684
=1/38 0.0263157894736842
=1/51 0.0196078431372549
=1/53 0.0188679245283019
=1/57 0.0175438596491228
=1/69 0.0144927536231884
=1/76 0.0131578947368421
=1/85 0.0117647058823529
=1/87 0.0114942528735632
=1/95 0.0105263157894737
=2/17 0.117647058823529
=2/19 0.105263157894737
=2/23 0.0869565217391304
=2/29 0.0689655172413793
=2/34 0.0588235294117647
=2/38 0.0526315789473684
=2/47 0.0425531914893617
=2/51 0.0392156862745098
=2/57 0.0350877192982456
=2/61 0.0327868852459016
=2/76 0.0263157894736842
=2/83 0.0240963855421687
=2/95 0.0210526315789474
=3/17 0.176470588235294
=3/19 0.157894736842105
=3/29 0.103448275862069
=3/34 0.0882352941176471
=3/38 0.0789473684210526
=3/46 0.0652173913043478
=3/51 0.0588235294117647
=3/57 0.0526315789473684
=3/61 0.0491803278688525
=3/76 0.0394736842105263
=3/85 0.0352941176470588
=3/95 0.0315789473684211
=3/97 0.0309278350515464
=3/98 0.0306122448979592
=4/19 0.210526315789474
=4/34 0.117647058823529
=4/38 0.105263157894737
=4/46 0.0869565217391304
=4/47 0.0851063829787234
=4/51 0.0784313725490196
=4/53 0.0754716981132075
=4/57 0.0701754385964912
=4/58 0.0689655172413793
=4/68 0.0588235294117647
=4/76 0.0526315789473684
=4/94 0.0425531914893617
=4/95 0.0421052631578947
=5/19 0.263157894736842
=5/38 0.131578947368421
=5/51 0.0980392156862745
=5/61 0.0819672131147541
=5/69 0.072463768115942
=5/76 0.0657894736842105
=5/83 0.0602409638554217
=5/85 0.0588235294117647
=5/94 0.0531914893617021
=5/95 0.0526315789473684
=6/17 0.352941176470588
=6/19 0.315789473684211
=6/29 0.206896551724138
=6/34 0.176470588235294
=6/38 0.157894736842105
=6/51 0.117647058823529
=6/53 0.113207547169811
=6/57 0.105263157894737
=6/58 0.103448275862069
=6/68 0.0882352941176471
=6/69 0.0869565217391304
=6/76 0.0789473684210526
=6/85 0.0705882352941176
=6/87 0.0689655172413793
=6/92 0.0652173913043478
=6/95 0.0631578947368421
=7/19 0.368421052631579
=7/34 0.205882352941176
=7/38 0.184210526315789
=7/51 0.137254901960784
=7/53 0.132075471698113
=7/76 0.0921052631578947
=7/85 0.0823529411764706
=7/89 0.0786516853932584
=7/97 0.0721649484536082
=8/19 0.421052631578947
=8/38 0.210526315789474
=8/51 0.156862745098039
=8/68 0.117647058823529
=8/76 0.105263157894737
=8/83 0.0963855421686747
=8/85 0.0941176470588235
=8/92 0.0869565217391304
=8/94 0.0851063829787234
=8/95 0.0842105263157895
=9/34 0.264705882352941
=9/51 0.176470588235294
=9/53 0.169811320754717
=9/57 0.157894736842105
=9/76 0.118421052631579
=9/85 0.105882352941176
=9/87 0.103448275862069
=9/95 0.0947368421052632
=9/97 0.0927835051546392

bluexuemei · 发表于 2011-9-13 23:41

香川群子发表于 2011-9-13 23:33
最后，再来一些除法的：
a=1-9, b=3-99,计算结果符合条件的有：

群子，这些东西，你是用什么算出来的？

香川群子 · 发表于 2011-9-13 23:45

本帖最后由香川群子于 2011-9-13 23:49 编辑

其中最佳是下面3个：

=5/19       0.263157894 736842       到第11位时数字全了
=1/38       0.0263157894 736842       到第12位时数字全了
=1/19       0.05263157894 73684       到第13位时数字全了

=1/17    0.05882352941176    47 则要到第16位时数字才出全。

香川群子 · 发表于 2011-9-13 23:56

bluexuemei 发表于 2011-9-13 23:41
群子，这些东西，你是用什么算出来的？

自己用VBA编写几行代码。

除法简单，用Excel自己的计算结果，加上一个判断函数就可以啦。
Function CK(N) '检查是否含有0-9
For k = 0 To 9
      If InStr(N, k) = 0 Then Exit Function
Next
CK = 1
End Function

Function CHK(N) '检查到第几个数位才能找齐0-9(位数含小数点)
C = "0123456789"
For i = 1 To Len(N)
      C = Replace(C, Mid(N, i, 1), "")
      If C = "" Then CHK = i: Exit For
Next
End Function

==============

至于乘法，就比较麻烦一点。
需要长数位运算的自定义函数了。

下面这个，是计算整数乘法结果的自定义函数：

Function TM2(Na, Nb) '只能处理正整数的乘法
l = 6
If Na = 0 Or Nb = 0 Then TM2 = "0": Exit Function
la = (Len(Na) - 1) \ l
ReDim a(la)
For i = 0 To la - 1
a(i) = Mid(Na, Len(Na) - l * i - l + 1, l)
Next
a(la) = Mid(Na, 1, Len(Na) - l * la)
lb = (Len(Nb) - 1) \ l
ReDim B(lb)
For i = 0 To lb - 1
B(i) = Mid(Nb, Len(Nb) - l * i - l + 1, l)
Next
B(lb) = Mid(Nb, 1, Len(Nb) - l * lb)
ReDim C(la + lb)
For i = 0 To la
For j = 0 To lb
C(la - i + lb - j) = C(la - i + lb - j) + a(i) * B(j)
Next
Next
For i = la + lb To 1 Step -1
If Len(C(i)) > l Then
If i = 1 Then
C(0) = C(0) + Left(C(i), Len(C(i)) - l)
If Len(C(0)) > l Then C(0) = Left(C(0), Len(C(0)) - l) & Right(C(0), l)
Else
C(i - 1) = Format(C(i - 1) + Left(C(i), Len(C(i)) - l), String(l, "0"))
End If
C(i) = Right(C(i), l)
Else
C(i) = Format(C(i), String(l, "0"))
End If
Next
TM2 = Join(C, "")
End Function

复制代码

在此基础上，可以写一个简单的幂运算函数：
Function MT(N, a)
MT = 1
For i = 1 To a
MT = TM2(MT, N)
Next
End Function

这样就可以计算N的a次幂了。

呵呵。

bluexuemei · 发表于 2011-9-14 00:00

香川群子发表于 2011-9-13 23:56
自己用VBA编写几行代码。

除法简单，用Excel自己的计算结果，加上一个判断函数就可以啦。

多谢群子,你是研究型人才,以后多指教!

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[求助] 那一个数的多少次方后包含有1234567890这10个数字

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