The following fractions are written on the board 1/n, 2/(n-1), 3/(n-2), ... , n/1 where n is a natural number. Alice calculated the differences of the neighboring fractions in this row and found among them 10000 fractions of type 1/k (with natural k). Prove that he can find even 5000 more of such these differences.
(In reply to
A boy named Alice by Math Man)
Indeed n=30030 = 2*3*5*7*11*13 produces a list of 62 although lower than 39260 = 2*3*5*7*11*17, though this lower number is not found in OEIS A093449. Perhaps all that is needed is a given number of distinct prime factors, all producing the same number of unit fraction differences
for 30030:
715/29315 714/29316 1/41 17/698 1/28618
1365/28665 1364/28666 1/21 62/1303 1/27363
1716/28314 1715/28315 2/33 49/809 1/26697
2080/27950 2079/27951 16/215 9/121 1/26015
2640/27390 2639/27391 8/83 29/301 1/24983
2926/27104 2925/27105 19/176 15/139 1/24464
3081/26949 3080/26950 79/691 4/35 1/24185
4005/26025 4004/26026 267/1735 2/13 1/22555
4291/25739 4290/25740 613/3677 1/6 1/22062
5005/25025 5004/25026 1/5 834/4171 1/20855
6006/24024 6005/24025 1/4 1201/4805 1/19220
6370/23660 6369/23661 7/26 193/717 1/18642
6721/23309 6720/23310 47/163 32/111 1/18093
6930/23100 6929/23101 3/10 533/1777 1/17770
7371/22659 7370/22660 27/83 67/206 1/17098
7645/22385 7644/22386 139/407 14/41 1/16687
8086/21944 8085/21945 311/844 7/19 1/16036
8295/21735 8294/21736 79/207 29/76 1/15732
8646/21384 8645/21385 131/324 19/47 1/15228
9010/21020 9009/21021 901/2102 3/7 1/14714
10011/20019 10010/20020 3337/6673 1/2 1/13346
10725/19305 10724/19306 5/9 766/1379 1/12411
11011/19019 11010/19020 11/19 367/634 1/12046
11935/18095 11934/18096 31/47 153/232 1/10904
12090/17940 12089/17941 31/46 157/233 1/10718
12376/17654 12375/17655 68/97 75/107 1/10379
12936/17094 12935/17095 28/37 199/263 1/9731
13300/16730 13299/16731 190/239 31/39 1/9321
13651/16379 13650/16380 1241/1489 5/6 1/8934
14301/15729 14300/15730 681/749 10/11 1/8239
15015/15015 15014/15016 1 7507/7508 1/7508
15016/15014 15015/15015 7508/7507 1 1/7507
15730/14300 15729/14301 11/10 749/681 1/6810
16380/13650 16379/13651 6/5 1489/1241 1/6205
16731/13299 16730/13300 39/31 239/190 1/5890
17095/12935 17094/12936 263/199 37/28 1/5572
17655/12375 17654/12376 107/75 97/68 1/5100
17941/12089 17940/12090 233/157 46/31 1/4867
18096/11934 18095/11935 232/153 47/31 1/4743
19020/11010 19019/11011 634/367 19/11 1/4037
19306/10724 19305/10725 1379/766 9/5 1/3830
20020/10010 20019/10011 2 6673/3337 1/3337
21021/9009 21020/9010 7/3 2102/901 1/2703
21385/8645 21384/8646 47/19 324/131 1/2489
21736/8294 21735/8295 76/29 207/79 1/2291
21945/8085 21944/8086 19/7 844/311 1/2177
22386/7644 22385/7645 41/14 407/139 1/1946
22660/7370 22659/7371 206/67 83/27 1/1809
23101/6929 23100/6930 1777/533 10/3 1/1599
23310/6720 23309/6721 111/32 163/47 1/1504
23661/6369 23660/6370 717/193 26/7 1/1351
24025/6005 24024/6006 4805/1201 4 1/1201
25026/5004 25025/5005 4171/834 5 1/834
25740/4290 25739/4291 6 3677/613 1/613
26026/4004 26025/4005 13/2 1735/267 1/534
26950/3080 26949/3081 35/4 691/79 1/316
27105/2925 27104/2926 139/15 176/19 1/285
27391/2639 27390/2640 301/29 83/8 1/232
27951/2079 27950/2080 121/9 215/16 1/144
28315/1715 28314/1716 809/49 33/2 1/98
28666/1364 28665/1365 1303/62 21 1/62
29316/714 29315/715 698/17 41 1/17
for example 165 has 3 prime factors, 3*5*11, and gives 6 = 2^3-2 unit fraction differences.
45/120 44/121 3/8 4/11 1/88
55/110 54/111 1/2 18/37 1/74
66/99 65/100 2/3 13/20 1/60
100/65 99/66 20/13 3/2 1/26
111/54 110/55 37/18 2 1/18
121/44 120/45 11/4 8/3 1/12
(for these, the n values quoted are the constant sum of numerator and denominator, not the top of the range of numerator (or denominator)).
So of course the smallest in each case is the primorial to which Math Man refers.
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Posted by Charlie
on 2021-02-21 14:07:43 |
re: A boy named Alice
