A number n for which the number of digits in its prime factorization is smaller than its number of digits. more
The first 600 frugal numbers :
125,
128,
243,
256,
343,
512,
625,
729,
1024,
1029,
1215,
1250,
1280,
1331,
1369,
1458,
1536,
1681,
1701,
1715,
1792,
1849,
1875,
2048,
2187,
2197,
2209,
2401,
2560,
2809,
3125,
3481,
3584,
3645,
3721,
4096,
4374,
4375,
4489,
4802,
4913,
5041,
5103,
5329,
6241,
6250,
6561,
6859,
6889,
7203,
7921,
8192,
9375,
9409,
10000,
10082,
10112,
10125,
10201,
10206,
10240,
10368,
10375,
10443,
10449,
10496,
10609,
10624,
10625,
10633,
10658,
10752,
10935,
10976,
10985,
11008,
11045,
11125,
11163,
11392,
11421,
11449,
11664,
11767,
11776,
11875,
11881,
11907,
12005,
12032,
12125,
12167,
12288,
12393,
12416,
12482,
12500,
12544,
12691,
12769,
12800,
12879,
12943,
13122,
13125,
13467,
13568,
13718,
13778,
13824,
13851,
14045,
14063,
14336,
14337,
14375,
14406,
14641,
14739,
14749,
14823,
14848,
15104,
15123,
15309,
15379,
15463,
15552,
15616,
15625,
15842,
15872,
15987,
16000,
16121,
16129,
16281,
16384,
16767,
16807,
16875,
17152,
17161,
17253,
17405,
17496,
17739,
17920,
18125,
18176,
18179,
18225,
18605,
18688,
18723,
18750,
18769,
18818,
18944,
19197,
19208,
19321,
19375,
19663,
19683,
20000,
20169,
20224,
20237,
20480,
20577,
20667,
20736,
20923,
20992,
21141,
21248,
21609,
21627,
21870,
21875,
21952,
22016,
22201,
22445,
22599,
22784,
22801,
22981,
23125,
23328,
23571,
23763,
24010,
24057,
24064,
24334,
24353,
24367,
24389,
24565,
24576,
24649,
24832,
25000,
25039,
25088,
25205,
25515,
25625,
26047,
26244,
26411,
26569,
26645,
26875,
26973,
27097,
27136,
27783,
27889,
28125,
28227,
28431,
28469,
28561,
28672,
29282,
29375,
29791,
29889,
29929,
30208,
30375,
30527,
30618,
30625,
31104,
31205,
31213,
31232,
31250,
31347,
31423,
32000,
32041,
32761,
32768,
32805,
33125,
33271,
33614,
34263,
34295,
34304,
34375,
34391,
34445,
34992,
35287,
35721,
36015,
36352,
36481,
36501,
36875,
37179,
37249,
37303,
37376,
38125,
38416,
38637,
38809,
39366,
39601,
39605,
40000,
40448,
40625,
40817,
40960,
41472,
41553,
41875,
42496,
42875,
43011,
43687,
43750,
43904,
43923,
44375,
44469,
44521,
45568,
45619,
45625,
45927,
46656,
46875,
47045,
48013,
48223,
48778,
48843,
49152,
49375,
49664,
49729,
50000,
50301,
50421,
50625,
50653,
51529,
51759,
51875,
52441,
52488,
53125,
53217,
54289,
54675,
55223,
55447,
55625,
57121,
57122,
57344,
57591,
58081,
59049,
59375,
59582,
60025,
60507,
60625,
60835,
62208,
62500,
63001,
63423,
64000,
64827,
64881,
65536,
65610,
65625,
65863,
66049,
67228,
67797,
68921,
69169,
69629,
69984,
70713,
71875,
72171,
72361,
73167,
73205,
73441,
74431,
76545,
76729,
76832,
78125,
78732,
78961,
79507,
80000,
80089,
80919,
81920,
83349,
83521,
84035,
84375,
85169,
85293,
85683,
85849,
87808,
88837,
89373,
89667,
90625,
91125,
91854,
93312,
93750,
94041,
94249,
96721,
96875,
97969,
98304,
98415,
98441,
100000,
100352,
100359,
100467,
100489,
100499,
100602,
100625,
100736,
100842,
100845,
100864,
100949,
101120,
101125,
101185,
101250,
101251,
101306,
101331,
101375,
101632,
101817,
101875,
101888,
102016,
102303,
102400,
102487,
102541,
102625,
102656,
102789,
102875,
102885,
102912,
102973,
103058,
103125,
103173,
103243,
103247,
103259,
103335,
103375,
103499,
103518,
103552,
103625,
103680,
103750,
103761,
103808,
103823,
103933,
103936,
104375,
104615,
104704,
104733,
104875,
104882,
104960,
104976,
105088,
105125,
105149,
105219,
105301,
105344,
105625,
105705,
105728,
105856,
105861,
106097,
106112,
106240,
106250,
106434,
106496,
106625,
106673,
106677,
107125,
107163,
107264,
107359,
107375,
107392,
107584,
107648,
107649,
107653,
107776,
107811,
107875,
107909,
107911,
108032,
108045,
108125,
108135,
108151,
108241,
108544,
108578,
108621,
108731,
108864,
109056,
109091,
109107,
109184,
109312,
109350,
109375,
109443,
109503,
109561,
109625,
109696,
109744,
109760,
109952,
110079,
110080,
110125,
110224,
110336,
110375,
110446,
110464,
110473,
110592,
110625,
110848,
110875,
110889,
110894,
111005,
111051,
111104,
111250,
111392,
111537,
111747,
111875,
111909,
112000,
112023,
112128,
112225,
112256,
112384,
112509,
112627,
112768,
112789,
112847,
112903,
112995,
112999,
113024,
113125,
113375,
113408,
113481,
113533,
113536,
113569,
113627,
113875,
113920,
113967,
114005,
114176,
114242,
114244,
114375,
114453,
114688,
114875,
114905,
114944,
115169,
115182,
115248,
115351,
115591,
115625,
115911,
115943,
116096,
116125,
116162,
116224,
116397,
116427.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1918207 values, from 125 to 999999189).
n\r | 0 | 1 |
2 | 581431 | 1336776 | 2 |
3 | 633268 | 643261 | 641678 | 3 |
4 | 418729 | 669539 | 162702 | 667237 | 4 |
5 | 495408 | 355547 | 355691 | 355831 | 355730 | 5 |
6 | 130690 | 418915 | 226395 | 502578 | 224346 | 415283 | 6 |
7 | 441964 | 246424 | 246490 | 245717 | 246497 | 245462 | 245653 | 7 |
8 | 398970 | 336019 | 82486 | 334800 | 19759 | 333520 | 80216 | 332437 | 8 |
9 | 493562 | 214409 | 213961 | 70532 | 214381 | 213923 | 69174 | 214471 | 213794 | 9 |
10 | 109761 | 237970 | 118236 | 237580 | 117606 | 385647 | 117577 | 237455 | 118251 | 238124 | 10 |
11 | 55827 | 186581 | 185887 | 186641 | 186617 | 186564 | 185780 | 185865 | 185888 | 186670 | 185887 |
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.