A number such that the sum obtained inserting a + among its digit in any position gives a prime. more
The magnanimous numbers up to 10
15 :
11,
12,
14,
16,
20,
21,
23,
25,
29,
30,
32,
34,
38,
41,
43,
47,
49,
50,
52,
56,
58,
61,
65,
67,
70,
74,
76,
83,
85,
89,
92,
94,
98,
101,
110,
112,
116,
118,
130,
136,
152,
158,
170,
172,
203,
209,
221,
227,
229,
245,
265,
281,
310,
316,
334,
338,
356,
358,
370,
376,
394,
398,
401,
403,
407,
425,
443,
449,
467,
485,
512,
518,
536,
538,
554,
556,
574,
592,
598,
601,
607,
625,
647,
661,
665,
667,
683,
710,
712,
730,
736,
754,
772,
776,
790,
794,
803,
809,
821,
845,
863,
881,
889,
934,
938,
952,
958,
970,
974,
992,
994,
998,
1001,
1112,
1130,
1198,
1310,
1316,
1598,
1756,
1772,
1910,
1918,
1952,
1970,
1990,
2209,
2221,
2225,
2249,
2261,
2267,
2281,
2429,
2447,
2465,
2489,
2645,
2681,
2885,
3110,
3170,
3310,
3334,
3370,
3398,
3518,
3554,
3730,
3736,
3794,
3934,
3974,
4001,
4027,
4063,
4229,
4247,
4265,
4267,
4427,
4445,
4463,
4643,
4825,
4883,
5158,
5176,
5374,
5516,
5552,
5558,
5594,
5752,
5972,
5992,
6001,
6007,
6067,
6265,
6403,
6425,
6443,
6485,
6601,
6685,
6803,
6821,
7330,
7376,
7390,
7394,
7534,
7556,
7592,
7712,
7934,
7970,
8009,
8029,
8221,
8225,
8801,
8821,
9118,
9172,
9190,
9338,
9370,
9374,
9512,
9598,
9710,
9734,
9752,
9910,
11116,
11152,
11170,
11558,
11930,
13118,
13136,
13556,
15572,
15736,
15938,
15952,
17716,
17752,
17992,
19972,
20209,
20261,
20861,
22061,
22201,
22801,
22885,
24407,
26201,
26285,
26881,
28285,
28429,
31370,
31756,
33118,
33538,
33554,
35116,
35776,
37190,
37556,
37790,
37930,
39158,
39394,
40001,
40043,
40049,
40067,
40427,
40463,
40483,
42209,
42265,
44009,
44443,
44447,
46445,
48089,
48265,
51112,
53176,
53756,
53918,
55516,
55552,
55558,
55576,
55774,
57116,
57754,
60007,
60047,
60403,
60443,
60667,
62021,
62665,
64645,
66667,
66685,
68003,
68683,
71536,
71572,
71716,
71752,
73156,
75374,
75556,
77152,
77554,
79330,
79370,
80009,
80029,
80801,
80849,
82265,
82285,
82825,
82829,
84265,
86081,
86221,
88061,
88229,
88265,
88621,
91792,
93338,
93958,
93994,
99712,
99998,
111112,
111118,
111170,
111310,
113170,
115136,
115198,
115772,
117116,
119792,
135158,
139138,
151156,
151592,
159118,
177556,
193910,
199190,
200209,
200809,
220021,
220661,
222245,
224027,
226447,
226681,
228601,
282809,
282881,
282889,
311156,
319910,
331118,
333770,
333994,
335156,
339370,
351938,
359794,
371116,
373130,
393554,
399710,
400049,
404249,
408049,
408889,
424607,
440843,
464447,
484063,
484445,
486685,
488489,
515116,
533176,
551558,
559952,
595592,
595598,
600881,
602081,
626261,
628601,
644485,
684425,
686285,
711512,
719710,
753316,
755156,
773554,
777712,
777776,
799394,
799712,
800483,
802061,
802081,
804863,
806021,
806483,
806681,
822265,
864883,
888485,
888601,
888643,
911390,
911518,
915752,
931130,
975772,
979592,
991118,
999994,
1115756,
1137770,
1191518,
1197370,
1353136,
1379930,
1533736,
1593538,
1711576,
1791110,
1795912,
1915972,
1951958,
2000221,
2008829,
2442485,
2604067,
2606647,
2664425,
2666021,
2828809,
2862445,
3155116,
3171710,
3193198,
3195338,
3195398,
3315358,
3373336,
3573716,
3737534,
3751576,
3939118,
4000483,
4408603,
4468865,
4488245,
4644407,
5115736,
5357776,
5551376,
5579774,
5731136,
5759594,
5959774,
6462667,
6600227,
6600443,
6608081,
6640063,
6640643,
6824665,
6864485,
6866683,
7113710,
7133110,
7139390,
7153336,
7159172,
7311170,
7351376,
7719370,
7959934,
7979534,
8044009,
8068201,
8608081,
8844449,
9171170,
9777910,
9959374,
11771992,
13913170,
15177112,
17115116,
19337170,
19713130,
20266681,
22086821,
22600601,
22862885,
26428645,
28862465,
33939518,
37959994,
40866083,
44866043,
48606043,
48804809,
51137776,
51513118,
53151376,
53775934,
59593574,
60402247,
60860603,
62202281,
64622665,
66864625,
66886483,
71553536,
77917592,
82486825,
86842265,
91959398,
95559998,
117711170,
222866845,
228440489,
244064027,
280422829,
331111958,
400044049,
460040803,
511151552,
593559374,
606202627,
608844043,
622622801,
622888465,
773719910,
844460063,
882428665,
995955112,
1777137770,
2240064227,
2444402809,
5753779594,
6464886245,
9151995592,
22226226625,
31993717930,
39393115598,
46884486265,
84448000009,
5391391551358,
97393713331910.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 561 values, from 11 to 97393713331910).
n\r | 0 | 1 |
2 | 308 | 253 | 2 |
3 | 3 | 263 | 295 | 3 |
4 | 127 | 170 | 181 | 83 | 4 |
5 | 135 | 129 | 96 | 106 | 95 | 5 |
6 | 2 | 108 | 151 | 1 | 155 | 144 | 6 |
7 | 67 | 90 | 77 | 75 | 80 | 78 | 94 | 7 |
8 | 77 | 103 | 76 | 57 | 50 | 67 | 105 | 26 | 8 |
9 | 0 | 92 | 88 | 3 | 80 | 98 | 0 | 91 | 109 | 9 |
10 | 71 | 62 | 59 | 45 | 50 | 64 | 67 | 37 | 61 | 45 | 10 |
11 | 6 | 69 | 50 | 48 | 45 | 50 | 63 | 48 | 58 | 59 | 65 |
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.