A number which can be expressed as a sum of positive powers of its digits. more
The first 600 d-powerful numbers :
1,
2,
3,
4,
5,
6,
7,
8,
9,
24,
43,
63,
89,
132,
135,
153,
175,
209,
224,
226,
262,
264,
267,
283,
332,
333,
334,
357,
370,
371,
372,
373,
374,
375,
376,
377,
378,
379,
407,
445,
463,
518,
598,
629,
739,
794,
849,
935,
994,
1034,
1073,
1074,
1234,
1255,
1306,
1323,
1326,
1349,
1364,
1386,
1498,
1542,
1634,
1672,
1676,
1765,
1836,
2048,
2062,
2063,
2064,
2083,
2114,
2130,
2131,
2132,
2133,
2134,
2135,
2136,
2137,
2138,
2139,
2179,
2194,
2203,
2205,
2223,
2224,
2225,
2227,
2240,
2241,
2242,
2243,
2244,
2245,
2246,
2247,
2248,
2249,
2263,
2264,
2283,
2314,
2315,
2332,
2333,
2334,
2335,
2336,
2352,
2353,
2355,
2356,
2372,
2373,
2376,
2379,
2394,
2395,
2396,
2407,
2423,
2427,
2443,
2445,
2449,
2464,
2465,
2483,
2517,
2532,
2536,
2537,
2554,
2572,
2574,
2577,
2597,
2607,
2623,
2626,
2627,
2628,
2645,
2664,
2733,
2735,
2737,
2738,
2739,
2753,
2757,
2773,
2792,
2793,
2794,
2803,
2824,
2843,
2849,
2863,
2864,
2932,
2933,
2934,
2939,
2953,
2973,
2994,
3145,
3147,
3165,
3167,
3212,
3214,
3215,
3234,
3235,
3236,
3237,
3238,
3252,
3254,
3255,
3257,
3274,
3292,
3294,
3295,
3298,
3324,
3342,
3345,
3362,
3385,
3432,
3435,
3436,
3437,
3453,
3454,
3457,
3474,
3475,
3476,
3477,
3492,
3496,
3523,
3525,
3527,
3543,
3566,
3569,
3612,
3635,
3652,
3657,
3674,
3678,
3692,
3706,
3723,
3725,
3726,
3746,
3749,
3764,
3765,
3786,
3856,
3940,
3941,
3942,
3943,
3944,
3945,
3946,
3947,
3948,
3949,
4114,
4132,
4133,
4136,
4150,
4151,
4152,
4153,
4154,
4155,
4156,
4157,
4158,
4159,
4173,
4194,
4209,
4224,
4225,
4228,
4240,
4241,
4242,
4243,
4244,
4245,
4246,
4247,
4248,
4249,
4260,
4261,
4262,
4263,
4264,
4265,
4266,
4267,
4268,
4269,
4285,
4288,
4316,
4332,
4339,
4352,
4353,
4355,
4373,
4396,
4397,
4403,
4405,
4424,
4443,
4447,
4449,
4469,
4485,
4572,
4573,
4574,
4599,
4622,
4623,
4624,
4644,
4648,
4649,
4685,
4732,
4733,
4735,
4736,
4738,
4826,
4849,
4869,
4932,
4933,
4950,
4951,
4952,
4953,
4954,
4955,
4956,
4957,
4958,
4959,
5182,
5232,
5236,
5258,
5259,
5276,
5320,
5321,
5322,
5323,
5324,
5325,
5326,
5327,
5328,
5329,
5342,
5343,
5344,
5364,
5366,
5367,
5384,
5432,
5453,
5547,
5562,
5567,
5632,
5653,
5657,
5746,
5766,
5832,
5833,
5858,
5873,
5877,
5902,
6244,
6247,
6249,
6423,
6424,
6443,
6579,
6603,
6649,
6669,
6714,
6732,
6792,
6793,
6794,
6849,
6932,
6934,
6935,
6937,
6939,
6972,
6974,
6976,
7032,
7123,
7234,
7279,
7296,
7299,
7323,
7329,
7343,
7363,
7432,
7582,
7632,
7639,
7836,
7906,
7928,
7944,
7962,
7964,
8200,
8201,
8202,
8203,
8204,
8205,
8206,
8207,
8208,
8209,
8223,
8224,
8225,
8228,
8243,
8245,
8262,
8263,
8283,
8288,
8289,
8316,
8352,
8376,
8396,
8443,
8462,
8536,
8623,
8647,
8667,
8712,
8756,
8792,
8825,
8864,
8932,
8970,
8971,
8972,
8973,
8974,
8975,
8976,
8977,
8978,
8979,
9122,
9234,
9236,
9237,
9272,
9273,
9274,
9324,
9342,
9347,
9362,
9385,
9474,
9478,
9493,
9654,
9693,
9697,
9723,
9725,
9947,
9963,
10322,
10693,
10694,
10722,
10723,
12298,
12324,
12343,
12345,
12347,
12367,
12415,
12432,
12434,
12495,
12542,
12562,
12615,
12634,
12695,
12723,
12854,
13132,
13133,
13173,
13332,
13337,
13373,
13393,
13443,
13532,
13933,
13978,
14276,
14295,
14346,
14362,
14363,
14369,
14697,
14722,
14726,
14786,
15422,
15423,
15686,
15822,
15882,
16213,
16234,
16274,
16293,
16346,
16363,
16430,
16431,
16432,
16433,
16434,
16435,
16436,
16437,
16438,
16439,
16453,
16470,
16471,
16472,
16473,
16474,
16475,
16476,
16477,
16478,
16479,
16494,
16524,
16543,
16634,
16729,
16742,
16743,
16857,
16927,
16942,
16984,
17026,
17133,
17157,
17224,
17240,
17241,
17242,
17243,
17244,
17245,
17246,
17247,
17248,
17249,
17287,
17332,
17334,
17338,
17352,
17357,
17372,
17376,
17378,
17407,
17424,
17443,
17445,
17448,
17463.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 374482 values, from 1 to 9999865).
n\r | 0 | 1 |
2 | 189378 | 185104 | 2 |
3 | 125018 | 124479 | 124985 | 3 |
4 | 91773 | 90882 | 97605 | 94222 | 4 |
5 | 49506 | 52972 | 101288 | 88141 | 82575 | 5 |
6 | 63379 | 61615 | 63135 | 61639 | 62864 | 61850 | 6 |
7 | 53759 | 53403 | 53449 | 53254 | 53408 | 53500 | 53709 | 7 |
8 | 45273 | 45133 | 51557 | 48648 | 46500 | 45749 | 46048 | 45574 | 8 |
9 | 41217 | 42211 | 40346 | 42594 | 40669 | 42422 | 41207 | 41599 | 42217 | 9 |
10 | 8972 | 8973 | 58278 | 56847 | 46835 | 40534 | 43999 | 43010 | 31294 | 35740 | 10 |
11 | 35929 | 32173 | 35531 | 33205 | 34580 | 34137 | 33389 | 35214 | 32379 | 35728 | 32217 |
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.