A number n such that the arithmetic mean of its divisors is an integer number. more
The first 600 arithmetic numbers :
1,
3,
5,
6,
7,
11,
13,
14,
15,
17,
19,
20,
21,
22,
23,
27,
29,
30,
31,
33,
35,
37,
38,
39,
41,
42,
43,
44,
45,
46,
47,
49,
51,
53,
54,
55,
56,
57,
59,
60,
61,
62,
65,
66,
67,
68,
69,
70,
71,
73,
77,
78,
79,
83,
85,
86,
87,
89,
91,
92,
93,
94,
95,
96,
97,
99,
101,
102,
103,
105,
107,
109,
110,
111,
113,
114,
115,
116,
118,
119,
123,
125,
126,
127,
129,
131,
132,
133,
134,
135,
137,
138,
139,
140,
141,
142,
143,
145,
147,
149,
150,
151,
153,
154,
155,
157,
158,
159,
161,
163,
164,
165,
166,
167,
168,
169,
173,
174,
177,
179,
181,
182,
183,
184,
185,
186,
187,
188,
189,
190,
191,
193,
195,
197,
198,
199,
201,
203,
204,
205,
206,
207,
209,
210,
211,
212,
213,
214,
215,
217,
219,
220,
221,
222,
223,
224,
227,
229,
230,
231,
233,
235,
236,
237,
238,
239,
241,
245,
246,
247,
248,
249,
251,
253,
254,
255,
257,
258,
259,
260,
261,
262,
263,
264,
265,
266,
267,
269,
270,
271,
273,
275,
276,
277,
278,
280,
281,
282,
283,
284,
285,
286,
287,
291,
293,
294,
295,
297,
299,
301,
302,
303,
304,
305,
307,
308,
309,
310,
311,
313,
315,
317,
318,
319,
321,
322,
323,
326,
327,
329,
330,
331,
332,
334,
335,
337,
339,
340,
341,
342,
343,
345,
347,
348,
349,
350,
351,
352,
353,
354,
355,
356,
357,
358,
359,
361,
365,
366,
367,
369,
371,
373,
374,
375,
376,
377,
378,
379,
380,
381,
382,
383,
385,
389,
390,
391,
393,
395,
397,
398,
399,
401,
402,
403,
404,
406,
407,
409,
411,
413,
414,
415,
417,
418,
419,
420,
421,
422,
423,
425,
426,
427,
428,
429,
430,
431,
432,
433,
434,
435,
437,
438,
439,
443,
445,
446,
447,
449,
451,
452,
453,
454,
455,
456,
457,
459,
460,
461,
462,
463,
464,
465,
467,
469,
470,
471,
473,
474,
476,
477,
478,
479,
480,
481,
483,
485,
486,
487,
489,
491,
492,
493,
494,
495,
497,
498,
499,
500,
501,
502,
503,
504,
505,
506,
507,
509,
510,
511,
513,
515,
517,
518,
519,
521,
523,
524,
526,
527,
531,
533,
534,
535,
537,
539,
540,
541,
542,
543,
545,
546,
547,
548,
550,
551,
552,
553,
555,
557,
558,
559,
561,
563,
564,
565,
566,
568,
569,
570,
571,
572,
573,
574,
575,
577,
579,
580,
581,
582,
583,
585,
587,
589,
590,
591,
593,
594,
595,
596,
597,
598,
599,
601,
602,
605,
606,
607,
608,
609,
611,
612,
613,
614,
615,
616,
617,
618,
619,
620,
621,
622,
623,
627,
629,
630,
631,
632,
633,
635,
636,
637,
638,
639,
641,
642,
643,
644,
645,
646,
647,
649,
651,
653,
654,
655,
658,
659,
660,
661,
662,
663,
665,
667,
668,
669,
670,
671,
672,
673,
677,
678,
679,
681,
682,
683,
685,
686,
687,
689,
690,
691,
692,
693,
694,
695,
697,
699,
701,
702,
703,
705,
707,
708,
709,
710,
713,
714,
715,
716,
717,
718,
719,
721,
723,
725,
726,
727,
728,
731,
733,
734,
735,
736,
737,
739,
740,
741,
742,
743,
744,
745,
747,
748,
749,
750,
751,
753,
755,
757,
758,
759,
761,
762,
763,
764,
765,
766,
767,
769,
770,
771,
773,
774,
777,
779,
780,
781,
782,
783,
785,
786,
787,
788,
789,
790,
791,
793,
795,
797,
798,
799,
801,
803,
805,
806,
807,
809,
811,
812,
813,
814,
815,
817,
820,
821,
822,
823,
824,
825,
826,
827.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 8262497 values, from 1 to 9999999).
n\r | 0 | 1 |
2 | 3478740 | 4783757 | 2 |
3 | 2761338 | 2627437 | 2873722 | 3 |
4 | 1489521 | 2391834 | 1989219 | 2391923 | 4 |
5 | 1650241 | 1644771 | 1652663 | 1650005 | 1664817 | 5 |
6 | 1254744 | 1613230 | 1209789 | 1506594 | 1014207 | 1663933 | 6 |
7 | 1268785 | 1164539 | 1165641 | 1164854 | 1165881 | 1165046 | 1167751 | 7 |
8 | 618090 | 1195861 | 750744 | 1195904 | 871431 | 1195973 | 1238475 | 1196019 | 8 |
9 | 746812 | 875802 | 957414 | 964453 | 875799 | 957714 | 1050073 | 875836 | 958594 | 9 |
10 | 719257 | 961668 | 690072 | 962896 | 699199 | 930984 | 683103 | 962591 | 687109 | 965618 | 10 |
11 | 827054 | 743342 | 743576 | 743543 | 743541 | 743503 | 743625 | 743566 | 743554 | 743533 | 743660 |
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.