Numbers divisible by the product of their digits. more
The first 600 Zuckerman numbers :
1,
2,
3,
4,
5,
6,
7,
8,
9,
11,
12,
15,
24,
36,
111,
112,
115,
128,
132,
135,
144,
175,
212,
216,
224,
312,
315,
384,
432,
612,
624,
672,
735,
816,
1111,
1112,
1113,
1115,
1116,
1131,
1176,
1184,
1197,
1212,
1296,
1311,
1332,
1344,
1416,
1575,
1715,
2112,
2144,
2232,
2916,
3111,
3132,
3168,
3171,
3276,
3312,
3915,
4112,
4224,
4416,
6144,
6624,
6912,
7112,
7119,
8112,
8832,
9315,
9612,
11111,
11112,
11115,
11133,
11172,
11212,
11232,
11313,
11328,
11331,
11424,
11664,
11711,
11872,
12112,
12128,
12132,
12216,
12224,
12288,
12312,
12432,
12712,
12768,
13113,
13131,
13212,
13248,
13272,
13311,
13824,
13932,
14112,
14144,
16128,
16224,
16416,
16632,
17115,
17136,
18112,
18432,
18816,
19116,
21112,
21132,
21184,
21216,
21248,
21312,
21672,
21728,
22112,
22144,
22176,
23112,
23328,
23424,
24112,
24192,
24912,
26112,
26136,
26712,
27132,
27216,
28416,
31113,
31131,
31212,
31311,
31488,
32112,
32172,
32616,
32832,
33111,
33264,
34272,
34992,
35175,
37212,
38112,
41112,
41232,
41616,
42112,
42192,
42336,
42432,
42624,
42912,
43632,
48128,
51975,
61344,
61824,
62112,
64128,
66312,
71232,
71316,
72128,
73332,
77175,
77616,
81216,
81312,
82112,
82944,
83232,
86112,
89712,
91728,
92232,
93312,
93744,
111111,
111112,
111115,
111132,
111135,
111168,
111195,
111212,
111216,
111264,
111272,
111312,
111315,
111424,
111432,
111612,
111915,
112112,
112116,
112128,
112224,
112416,
113112,
113115,
113184,
113292,
113319,
113328,
113616,
113715,
114112,
114192,
114312,
114432,
114624,
114816,
114912,
116112,
116172,
116316,
117264,
117432,
117824,
117936,
118112,
118144,
118272,
119115,
119232,
119313,
121112,
121116,
121344,
121392,
121716,
121932,
122112,
123264,
123732,
124112,
124416,
126144,
126432,
131112,
131115,
131139,
131328,
131424,
131712,
131733,
132192,
134112,
134136,
136224,
139212,
139968,
141112,
141184,
141192,
141216,
141312,
141824,
141912,
142112,
143112,
144128,
144384,
146112,
147112,
149112,
161112,
161136,
162432,
163296,
164112,
167328,
171115,
171612,
172116,
173313,
175175,
177135,
177184,
179424,
181248,
182112,
182784,
183168,
183312,
186624,
189216,
191115,
192132,
193212,
194112,
196128,
197316,
198144,
211112,
211116,
211176,
212112,
212224,
213192,
214112,
214144,
217112,
218112,
219132,
219744,
221112,
221184,
221232,
221616,
222112,
222192,
222912,
229392,
231264,
231336,
232128,
232416,
234432,
237132,
239112,
241472,
241632,
242112,
244224,
248832,
261216,
266112,
272832,
282112,
282624,
291168,
293112,
294192,
311112,
311115,
311328,
311424,
311472,
311931,
312336,
312732,
313173,
313416,
313632,
314112,
316116,
316224,
317184,
317331,
321192,
331317,
331344,
333315,
336312,
338112,
341112,
341712,
342144,
344736,
371112,
371175,
373212,
381312,
382464,
383616,
391311,
411112,
411192,
411216,
411264,
411312,
411648,
411712,
411912,
412112,
412416,
413112,
413184,
414112,
416112,
417312,
418112,
419112,
421632,
421888,
422112,
422144,
431112,
431136,
434112,
437472,
438912,
441216,
441312,
442112,
442176,
442368,
476112,
487424,
491112,
511175,
611112,
612864,
613116,
613332,
614112,
621216,
622272,
626112,
628224,
642816,
647136,
661248,
662112,
671328,
688128,
698112,
711312,
711424,
711872,
713664,
714112,
717115,
719712,
721112,
724416,
731115,
731472,
732312,
733131,
733824,
739935,
741216,
742112,
772632,
811296,
813312,
817712,
821248,
831312,
842112,
843264,
863136,
911115,
911331,
911736,
912384,
913113,
914112,
916272,
918144,
921132,
929232,
941112,
941184,
941472,
942192,
964224,
972216,
973728,
1111111,
1111112,
1111113,
1111115,
1111116,
1111117,
1111131,
1111173,
1111212,
1111311,
1111344,
1111416,
1111424,
1111712,
1111968,
1112112,
1112128,
1112224,
1112232,
1112272,
1112616,
1112832,
1113111,
1113264,
1113912,
1114112,
1114176,
1116216,
1116432,
1117116,
1117935,
1118112,
1119312,
1121112,
1121232,
1121616,
1122112,
1122192,
1122312,
1122336,
1122432,
1122624,
1122816,
1122912,
1123632,
1124112,
1124224,
1126272,
1127112,
1127196,
1128448,
1131111,
1131192,
1131264,
1131711,
1131732,
1132128,
1132224,
1132416,
1134144,
1134432,
1141112,
1141248,
1141632,
1142112,
1142144,
1143216,
1143744,
1161216,
1161972,
1162944,
1164288,
1166112,
1167264,
1168128,
1169316,
1171212,
1173312,
1177176,
1177911,
1181184,
1181376,
1181712,
1182112,
1182272,
1191132,
1191915,
1193184,
1193616,
1194912,
1197315,
1211112,
1211232,
1211328,
1211616,
1212112,
1212192,
1212224,
1212312,
1212912,
1213344,
1213824,
1214112,
1216128,
1217664,
1217916,
1218112,
1218432,
1218816,
1221112,
1221192,
1221216,
1221312,
1221472,
1221912,
1222112,
1222144,
1223112,
1223424,
1223712,
1226112,
1229112,
1229184,
1231272.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 21896 values, from 1 to 9993393711).
n\r | 0 | 1 |
2 | 21355 | 541 | 2 |
3 | 20506 | 698 | 692 | 3 |
4 | 21353 | 87 | 2 | 454 | 4 |
5 | 261 | 3793 | 12298 | 2069 | 3475 | 5 |
6 | 20001 | 20 | 676 | 505 | 678 | 16 | 6 |
7 | 10268 | 1976 | 1885 | 2005 | 1891 | 1924 | 1947 | 7 |
8 | 20742 | 50 | 1 | 191 | 611 | 37 | 1 | 263 | 8 |
9 | 17118 | 229 | 234 | 1632 | 240 | 229 | 1756 | 229 | 229 | 9 |
10 | 0 | 171 | 12273 | 61 | 3452 | 261 | 3622 | 25 | 2008 | 23 | 10 |
11 | 2099 | 1957 | 2063 | 1918 | 2083 | 1894 | 2016 | 1988 | 1955 | 2008 | 1915 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.