It was only in my previous post that I mentioned Friedman numbers, named after the former Associate Professor of Mathematics at Stetson University in DeLand, Florida. His name popped up again this morning when I was investigating the number associated with my diurnal age: 26124. Before discussing the mathematical property of this number, namely its "nudity", I'll include some biographical information about Erich Friedman that I found on his website:
My name is Erich Friedman. For 26 years, I was a Professor of Mathematics at Stetson University, located in DeLand, Florida. I retired in 2018 to spend more time on my other interests, including recreational mathematics, puzzles, trivia, and my girlfriend of 30 years. I was born in 1965 in West Lafayette, Indiana. I grew up in Indianapolis and went to North Central High School. I got my bachelor's degree from Rose-Hulman in 1987, and my Ph.D. from Cornell University in 1991, and have been at Stetson ever since.
There's more information on his website but that's enough for this post. Suffice to say that his website looks interesting with many links to mathematical topics. Eric Friedman is the author of OEIS A034838: numbers \(n\) that are divisible by every digit of \(n\). 26214 is a member of this sequence because 1, 2, 4 and 6 do indeed divide into it without remainder.
I was lead to this sequence by a link in Numbers Aplenty concerning what are colorfully called nude numbers, so called because such numbers expose some of their factors. The explanation on the Numbers Aplenty website runs like this:
Y.Katagiri calls a number nude if it is divisible by all of its digits (which should be nonzero) like \(672=6\cdot112=7\cdot96=2\cdot 336\). The number is called "nude" because it exposes some of its factors. There are only \(9039\) such numbers below one million, however there are infinite nude numbers since all repdigits are nude. The smallest nude number which contains all the odd digits is \(1117935\). Note that if a nude number contains a \(5\), then all the other digits must be odd. The smallest nude \(n\) which contains the maximal (8) number of distinct digits is \(1123449768\). The smallest triple of consecutive nontrivial nude numbers is \((1111, 1112, 1113)\). It is easy to see that there cannot be four consecutive nude numbers greater than 10.
The entry goes on to depict the smallest 3 × 3 magic square whose entries are nontrivial consecutive nude numbers (that is, not the numbers from 1 to 9). See Figure 1.
Figure 1
It's easy enough to generate all the nude numbers from 1 to 26124 using the SageMath code shown below (permalink to SageMathCell):
L=[]for n in [1..26124]:N=n.digits()OK=1for i in range(len(N)):if N[i]==0:OK=0breakelse:if n%N[i]!=0:OK=0breakif OK==1:L.append(n)print(L)print("The percentage of nude numbers up to",n,"is",numerical_approx(len(L)/n*100,digits=2))
The output tells us that approximately 2.9% of the numbers between 1 and 26124 are nude. Here is output:
[1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128, 132, 135, 144, 155, 162, 168, 175, 184, 212, 216, 222, 224, 244, 248, 264, 288, 312, 315, 324, 333, 336, 366, 384, 396, 412, 424, 432, 444, 448, 488, 515, 555, 612, 624, 636, 648, 666, 672, 728, 735, 777, 784, 816, 824, 848, 864, 888, 936, 999, 1111, 1112, 1113, 1115, 1116, 1122, 1124, 1128, 1131, 1144, 1155, 1164, 1176, 1184, 1197, 1212, 1222, 1224, 1236, 1244, 1248, 1266, 1288, 1296, 1311, 1326, 1332, 1335, 1344, 1362, 1368, 1395, 1412, 1416, 1424, 1444, 1448, 1464, 1488, 1515, 1555, 1575, 1626, 1632, 1644, 1662, 1692, 1715, 1722, 1764, 1771, 1824, 1848, 1888, 1926, 1935, 1944, 1962, 2112, 2122, 2124, 2128, 2136, 2144, 2166, 2184, 2196, 2212, 2222, 2224, 2226, 2232, 2244, 2248, 2262, 2288, 2316, 2322, 2328, 2364, 2412, 2424, 2436, 2444, 2448, 2488, 2616, 2622, 2664, 2688, 2744, 2772, 2824, 2832, 2848, 2888, 2916, 3111, 3126, 3132, 3135, 3144, 3162, 3168, 3171, 3195, 3216, 3222, 3264, 3276, 3288, 3312, 3315, 3324, 3333, 3336, 3339, 3366, 3384, 3393, 3432, 3444, 3492, 3555, 3612, 3624, 3636, 3648, 3666, 3717, 3816, 3864, 3888, 3915, 3924, 3933, 3996, 4112, 4116, 4124, 4128, 4144, 4164, 4172, 4184, 4212, 4224, 4236, 4244, 4248, 4288, 4332, 4344, 4368, 4392, 4412, 4416, 4424, 4444, 4448, 4464, 4488, 4632, 4644, 4824, 4848, 4872, 4888, 4896, 4932, 4968, 5115, 5155, 5355, 5515, 5535, 5555, 5775, 6126, 6132, 6144, 6162, 6168, 6192, 6216, 6222, 6264, 6288, 6312, 6324, 6336, 6366, 6384, 6432, 6444, 6612, 6624, 6636, 6648, 6666, 6696, 6762, 6816, 6864, 6888, 6912, 6966, 6984, 7112, 7119, 7175, 7224, 7266, 7371, 7448, 7476, 7644, 7728, 7777, 7784, 8112, 8128, 8136, 8144, 8184, 8224, 8232, 8248, 8288, 8328, 8424, 8448, 8488, 8496, 8616, 8664, 8688, 8736, 8824, 8832, 8848, 8888, 8928, 9126, 9135, 9144, 9162, 9216, 9288, 9315, 9324, 9333, 9396, 9432, 9612, 9648, 9666, 9864, 9936, 9999, 11111, 11112, 11115, 11122, 11124, 11128, 11133, 11136, 11144, 11155, 11166, 11172, 11184, 11196, 11212, 11222, 11224, 11226, 11232, 11244, 11248, 11262, 11288, 11313, 11316, 11322, 11328, 11331, 11355, 11364, 11412, 11424, 11436, 11444, 11448, 11488, 11515, 11535, 11555, 11616, 11622, 11664, 11676, 11688, 11711, 11824, 11832, 11848, 11872, 11888, 11916, 12112, 12122, 12124, 12126, 12128, 12132, 12144, 12162, 12168, 12184, 12212, 12216, 12222, 12224, 12244, 12248, 12264, 12288, 12312, 12324, 12336, 12366, 12384, 12412, 12424, 12432, 12444, 12448, 12488, 12492, 12612, 12624, 12636, 12648, 12666, 12712, 12726, 12768, 12816, 12824, 12848, 12864, 12888, 12924, 12996, 13113, 13116, 13122, 13128, 13131, 13155, 13164, 13212, 13224, 13236, 13248, 13266, 13272, 13311, 13326, 13332, 13335, 13344, 13362, 13368, 13377, 13392, 13416, 13464, 13488, 13515, 13626, 13632, 13644, 13662, 13713, 13755, 13776, 13797, 13824, 13848, 13896, 13932, 13968, 13995, 14112, 14124, 14128, 14136, 14144, 14184, 14212, 14224, 14232, 14244, 14248, 14288, 14292, 14316, 14328, 14364, 14412, 14424, 14436, 14444, 14448, 14488, 14616, 14664, 14688, 14728, 14784, 14824, 14832, 14848, 14888, 15115, 15135, 15155, 15315, 15515, 15555, 15575, 15715, 16116, 16122, 16128, 16164, 16212, 16224, 16236, 16248, 16266, 16326, 16332, 16344, 16362, 16368, 16416, 16464, 16488, 16626, 16632, 16644, 16662, 16716, 16824, 16848, 16992, 17115, 17122, 17136, 17171, 17199, 17248, 17262, 17444, 17472, 17535, 17717, 17724, 17766, 17955, 18112, 18128, 18144, 18168, 18184, 18216, 18224, 18248, 18264, 18288, 18312, 18336, 18384, 18424, 18432, 18448, 18488, 18624, 18648, 18816, 18824, 18848, 18864, 18872, 18888, 18936, 19116, 19224, 19296, 19332, 19368, 19395, 19692, 19719, 19926, 19935, 19944, 19962, 19971, 21112, 21122, 21124, 21126, 21128, 21132, 21144, 21162, 21168, 21184, 21212, 21216, 21222, 21224, 21244, 21248, 21264, 21288, 21312, 21324, 21336, 21366, 21384, 21412, 21424, 21432, 21444, 21448, 21488, 21492, 21612, 21624, 21636, 21648, 21666, 21672, 21728, 21784, 21816, 21824, 21848, 21864, 21888, 21924, 21996, 22112, 22116, 22122, 22124, 22128, 22144, 22164, 22176, 22184, 22212, 22222, 22224, 22236, 22244, 22248, 22266, 22288, 22326, 22332, 22344, 22362, 22368, 22392, 22412, 22416, 22424, 22444, 22448, 22464, 22488, 22626, 22632, 22644, 22662, 22722, 22764, 22824, 22848, 22888, 22896, 22932, 22968, 23112, 23124, 23136, 23166, 23184, 23226, 23232, 23244, 23262, 23292, 23316, 23322, 23328, 23364, 23412, 23424, 23436, 23448, 23616, 23622, 23664, 23688, 23772, 23832, 23922, 24112, 24124, 24128, 24132, 24144, 24168, 24184, 24192, 24212, 24216, 24224, 24244, 24248, 24264, 24276, 24288, 24312, 24324, 24336, 24384, 24412, 24424, 24432, 24444, 24448, 24472, 24488, 24612, 24624, 24636, 24648, 24696, 24724, 24816, 24824, 24848, 24864, 24888, 24912, 24984, 26112, 26124]
The percentage of nude numbers up to 26124 is 2.9
Wolfram MathWorld has some auxiliary information:
Numbers in base-10 which are divisible by their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, ... (OEIS A034838). Numbers which are divisible by the sum of their digits are called Harshad numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (OEIS A005349). Numbers which are divisible by both their digits and the sum of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 111, 112, 126, 132, 135, 144, ... (OEIS A050104). Numbers which are equal to (i.e., not just divisible by) the product of their divisors and the sum of their divisors are called sum-product numbers and are given by 1, 135, 144, ... (OEIS A038369).
No comments:
Post a Comment