A number n whose representation can be decomposed into two parts, a and b, such that n is equal to the sum of the integers from a to b. more
The astonishing numbers up to 10
15 :
15,
27,
204,
216,
429,
1353,
1863,
3078,
3388,
3591,
7119,
20328,
21252,
21762,
23287,
23490,
78403,
133533,
178623,
1301613,
2002077,
2006118,
2077402,
2132532,
2177622,
2282148,
2732353,
3882813,
7103835,
13335333,
17016076,
17786223,
27377889,
32738728,
35639163,
124015749,
202272147,
202722352,
203872812,
207093834,
213325332,
217006075,
217776222,
227367888,
232728727,
235629162,
308725039,
347826603,
793039833,
1248851513,
1333353333,
1420855168,
1777862223,
2188066188,
2783074658,
2963077038,
3146385338,
3699393633,
4278092598,
5338103463,
7063119063,
9063134938,
12643159518,
15238175238,
17147185979,
20001201612,
20308625038,
20347726602,
21248751512,
21333253332,
21420755167,
21777762222,
22448213073,
23146285337,
23699293632,
23788219413,
27313235313,
29058242818,
34488264888,
36763273638,
38788281213,
43273297353,
50813322813,
53578331703,
57143342858,
62586359290,
71638385238,
77838402163,
84203418923,
86688425313,
93413442213,
96670450206,
100513459488,
112013486388,
112988488613,
119688503688,
125200515825,
133333533333,
141873551248,
153927575802,
155476578900,
171113609513,
177778622223,
187567640559,
194563653438,
204838672038,
221538701538,
227509711884,
232738720863,
244063740063,
272413786788,
277313794688,
289788814588,
313693851933,
320938863063,
327273872728,
340964893412,
342318895443,
356363916363,
363088926288,
370963937838,
396122974247,
398438977563,
413763999363,
646031138523,
944381377563,
2000123015748,
2000792039832,
2002187066187,
2002782074657,
2002962077037,
2004277092597,
2005337103462,
2007062119062,
2009062134937,
2012642159517,
2015237175237,
2017146185978,
2022447213072,
2023787219412,
2027312235312,
2029057242817,
2034487264887,
2036762273637,
2038787281212,
2043272297352,
2050812322812,
2053577331702,
2057142342857,
2062585359289,
2071637385237,
2077837402162,
2084202418922,
2086687425312,
2093412442212,
2096669450205,
2100512459487,
2112012486387,
2112987488612,
2119687503687,
2125199515824,
2133332533332,
2141872551247,
2153926575801,
2155475578899,
2171112609512,
2177777622222,
2187566640558,
2194562653437,
2204837672037,
2221537701537,
2227508711883,
2232737720862,
2244062740062,
2272412786787,
2277312794687,
2289787814587,
2313692851932,
2320937863062,
2327272872727,
2340963893411,
2342317895442,
2356362916362,
2363087926287,
2370962937837,
2396121974246,
2398437977562,
2413762999362,
6998303741858,
7056633822463,
7993384077463,
12468804995313,
13333335333333,
17777786222223,
20137626658114,
20768806448313,
21750636944938,
42043809179563,
73933812182538,
100316314199963,
100419814207323,
122092615674051,
133566816398708,
142391216935464,
200646021138522,
200944371377562,
207056623822462,
207993374077462,
213333325333332,
217777776222222,
220137616658113,
221750626944937,
228118821481188,
273131323531313,
315656325323438,
340493826316938,
387878828121213,
591485334899228,
610111335460488,
689693837775063,
709843838341563,
874688041834688,
891406343154063.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 223 values, from 15 to 891406343154063).
n\r | 0 | 1 |
2 | 111 | 112 | 2 |
3 | 157 | 37 | 29 | 3 |
4 | 54 | 52 | 57 | 60 | 4 |
5 | 8 | 8 | 84 | 108 | 15 | 5 |
6 | 77 | 19 | 16 | 80 | 18 | 13 | 6 |
7 | 88 | 34 | 16 | 19 | 19 | 21 | 26 | 7 |
8 | 25 | 25 | 28 | 28 | 29 | 27 | 29 | 32 | 8 |
9 | 97 | 10 | 11 | 23 | 20 | 10 | 37 | 7 | 8 | 9 |
10 | 3 | 4 | 50 | 60 | 6 | 5 | 4 | 34 | 48 | 9 | 10 |
11 | 100 | 8 | 14 | 9 | 22 | 16 | 13 | 11 | 13 | 8 | 9 |
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.