Sastry numbers

A number which gives a square when concatenated with its successor. more

The Sastry numbers up to 10¹⁵ :

183, 328, 528, 715, 6099, 13224, 40495, 106755, 453288, 2066115, 2975208, 22145328, 28027683, 110213248, 110667555, 147928995, 178838403, 226123528, 275074575, 333052608, 378698224, 445332888, 446245635, 518348515, 574930563, 647238399, 657515568, 734693880, 801777640, 826446280, 897506928, 898802499, 924910143, 1568473680, 3647681599, 4789624063, 8315420899, 13973312520, 16311962328, 20087347599, 22873345935, 27221172024, 30449526120, 35535821475, 39211524099, 80200635024, 85677207840, 94076078655, 111066675555, 444533328888, 547194595608, 859342532899, 1520909198095, 2312623569024, 2694674579715, 3721142026968, 8264462809920, 11080783971075, 18125673953535, 21459780540675, 28810375705728, 32977159102048, 44505160891528, 49649827129848, 52000863121368, 57550535077288, 92639768392899, 111106666755555, 116628474462408, 118202271352920, 119504810519103, 125229019411320, 129894351421315, 132681971063043, 138962394247423, 157175907513603, 159680479360800, 165594526329648, 167135574599055, 169717971697224, 178376694023055, 179382820426083, 190012386574095, 206611570247935, 208332508201368, 216247655238520, 218008184720899, 231964930624575, 235685153224255, 244031444160099, 256040666551120, 264736847238288, 268710175900624, 284181252744960, 286198935571599, 295463386801443, 297520661157024, 318204181579224, 332310638316240, 333683391047224, 345781149611395, 349491215237224, 351728379276835, 360479680159999, 364267571036800, 393409192262688, 404171240701320, 409077172321608, 417474769939599, 428115454167408, 430591127310243, 433610247875715, 444453333288888, 469656217907968, 472249051009635, 486761145354400, 503036096947263, 507508927276848, 510204081632655, 520733098205283, 525283753785840, 550784910234000, 555464752957503, 586365207364440, 588813227892643, 591715976331360, 607947176538175, 640384270017808, 653547066691855, 670434782384488, 673531954442275, 678706013980624, 684461783579328, 690841625906560, 718717107443715, 745403671829640, 750846279256899, 768939765623640, 772256418894399, 786704531939448, 790783556533224, 792900413637880, 820588019359999, 849086542230624, 883636862556720, 887192024930703, 893128849172808, 899729731503024, 907042176788208, 938942853643023, 946569584910399, 979151996880024, 996225356413048.

Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 153 values, from 183 to 996225356413048).

n\r	0	1
2	81	72	2
3	109	44	0	3
4	81	0	0	72	4
5	62	0	0	57	34	5
6	58	21	0	51	23	0	6
7	26	24	23	19	16	19	26	7
8	81	0	0	39	0	0	0	33	8
9	40	0	0	35	44	0	34	0	0	9
10	27	0	0	19	16	35	0	0	38	18	10
11	25	13	13	10	15	13	11	16	9	18	10

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.

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n\r	0	1
2	81	72	2
3	109	44	0	3
4	81	0	0	72	4
5	62	0	0	57	34	5
6	58	21	0	51	23	0	6
7	26	24	23	19	16	19	26	7
8	81	0	0	39	0	0	0	33	8
9	40	0	0	35	44	0	34	0	0	9
10	27	0	0	19	16	35	0	0	38	18	10
11	25	13	13	10	15	13	11	16	9	18	10

n\r	0	1
2	81	72	2
3	109	44	0	3
4	81	0	0	72	4
5	62	0	0	57	34	5
6	58	21	0	51	23	0	6
7	26	24	23	19	16	19	26	7
8	81	0	0	39	0	0	0	33	8
9	40	0	0	35	44	0	34	0	0	9
10	27	0	0	19	16	35	0	0	38	18	10
11	25	13	13	10	15	13	11	16	9	18	10

n\r	0	1
2	81	72	2
3	109	44	0	3
4	81	0	0	72	4
5	62	0	0	57	34	5
6	58	21	0	51	23	0	6
7	26	24	23	19	16	19	26	7
8	81	0	0	39	0	0	0	33	8
9	40	0	0	35	44	0	34	0	0	9
10	27	0	0	19	16	35	0	0	38	18	10
11	25	13	13	10	15	13	11	16	9	18	10