A number n such that there exist n integer numbers whose product is equal to their sum. more
The first 600 amenable numbers :
1,
5,
8,
9,
12,
13,
16,
17,
20,
21,
24,
25,
28,
29,
32,
33,
36,
37,
40,
41,
44,
45,
48,
49,
52,
53,
56,
57,
60,
61,
64,
65,
68,
69,
72,
73,
76,
77,
80,
81,
84,
85,
88,
89,
92,
93,
96,
97,
100,
101,
104,
105,
108,
109,
112,
113,
116,
117,
120,
121,
124,
125,
128,
129,
132,
133,
136,
137,
140,
141,
144,
145,
148,
149,
152,
153,
156,
157,
160,
161,
164,
165,
168,
169,
172,
173,
176,
177,
180,
181,
184,
185,
188,
189,
192,
193,
196,
197,
200,
201,
204,
205,
208,
209,
212,
213,
216,
217,
220,
221,
224,
225,
228,
229,
232,
233,
236,
237,
240,
241,
244,
245,
248,
249,
252,
253,
256,
257,
260,
261,
264,
265,
268,
269,
272,
273,
276,
277,
280,
281,
284,
285,
288,
289,
292,
293,
296,
297,
300,
301,
304,
305,
308,
309,
312,
313,
316,
317,
320,
321,
324,
325,
328,
329,
332,
333,
336,
337,
340,
341,
344,
345,
348,
349,
352,
353,
356,
357,
360,
361,
364,
365,
368,
369,
372,
373,
376,
377,
380,
381,
384,
385,
388,
389,
392,
393,
396,
397,
400,
401,
404,
405,
408,
409,
412,
413,
416,
417,
420,
421,
424,
425,
428,
429,
432,
433,
436,
437,
440,
441,
444,
445,
448,
449,
452,
453,
456,
457,
460,
461,
464,
465,
468,
469,
472,
473,
476,
477,
480,
481,
484,
485,
488,
489,
492,
493,
496,
497,
500,
501,
504,
505,
508,
509,
512,
513,
516,
517,
520,
521,
524,
525,
528,
529,
532,
533,
536,
537,
540,
541,
544,
545,
548,
549,
552,
553,
556,
557,
560,
561,
564,
565,
568,
569,
572,
573,
576,
577,
580,
581,
584,
585,
588,
589,
592,
593,
596,
597,
600,
601,
604,
605,
608,
609,
612,
613,
616,
617,
620,
621,
624,
625,
628,
629,
632,
633,
636,
637,
640,
641,
644,
645,
648,
649,
652,
653,
656,
657,
660,
661,
664,
665,
668,
669,
672,
673,
676,
677,
680,
681,
684,
685,
688,
689,
692,
693,
696,
697,
700,
701,
704,
705,
708,
709,
712,
713,
716,
717,
720,
721,
724,
725,
728,
729,
732,
733,
736,
737,
740,
741,
744,
745,
748,
749,
752,
753,
756,
757,
760,
761,
764,
765,
768,
769,
772,
773,
776,
777,
780,
781,
784,
785,
788,
789,
792,
793,
796,
797,
800,
801,
804,
805,
808,
809,
812,
813,
816,
817,
820,
821,
824,
825,
828,
829,
832,
833,
836,
837,
840,
841,
844,
845,
848,
849,
852,
853,
856,
857,
860,
861,
864,
865,
868,
869,
872,
873,
876,
877,
880,
881,
884,
885,
888,
889,
892,
893,
896,
897,
900,
901,
904,
905,
908,
909,
912,
913,
916,
917,
920,
921,
924,
925,
928,
929,
932,
933,
936,
937,
940,
941,
944,
945,
948,
949,
952,
953,
956,
957,
960,
961,
964,
965,
968,
969,
972,
973,
976,
977,
980,
981,
984,
985,
988,
989,
992,
993,
996,
997,
1000,
1001,
1004,
1005,
1008,
1009,
1012,
1013,
1016,
1017,
1020,
1021,
1024,
1025,
1028,
1029,
1032,
1033,
1036,
1037,
1040,
1041,
1044,
1045,
1048,
1049,
1052,
1053,
1056,
1057,
1060,
1061,
1064,
1065,
1068,
1069,
1072,
1073,
1076,
1077,
1080,
1081,
1084,
1085,
1088,
1089,
1092,
1093,
1096,
1097,
1100,
1101,
1104,
1105,
1108,
1109,
1112,
1113,
1116,
1117,
1120,
1121,
1124,
1125,
1128,
1129,
1132,
1133,
1136,
1137,
1140,
1141,
1144,
1145,
1148,
1149,
1152,
1153,
1156,
1157,
1160,
1161,
1164,
1165,
1168,
1169,
1172,
1173,
1176,
1177,
1180,
1181,
1184,
1185,
1188,
1189,
1192,
1193,
1196,
1197,
1200,
1201.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 4999999 values, from 1 to 10000000).
n\r | 0 | 1 |
2 | 2499999 | 2500000 | 2 |
3 | 1666666 | 1666667 | 1666666 | 3 |
4 | 2499999 | 2500000 | 0 | 0 | 4 |
5 | 1000000 | 1000000 | 1000000 | 1000000 | 999999 | 5 |
6 | 833333 | 833334 | 833333 | 833333 | 833333 | 833333 | 6 |
7 | 714285 | 714286 | 714286 | 714286 | 714284 | 714286 | 714286 | 7 |
8 | 1250000 | 1250000 | 0 | 0 | 1249999 | 1250000 | 0 | 0 | 8 |
9 | 555555 | 555556 | 555555 | 555556 | 555555 | 555555 | 555555 | 555556 | 555556 | 9 |
10 | 500000 | 500000 | 500000 | 500000 | 499999 | 500000 | 500000 | 500000 | 500000 | 500000 | 10 |
11 | 454544 | 454546 | 454546 | 454545 | 454544 | 454546 | 454546 | 454545 | 454545 | 454546 | 454546 |
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.