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astonishing numbers
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 1015 :
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 
2111112 2 
31573729 3 
454525760 4 
5888410815 5 
6771916801813 6 
788341619192126 7 
82525282829272932 8 
99710112320103778 9 
1034506065434489 10 
111008149221613111389

A pictorial representation of the table above
motab
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.