Pairs of consecutive numbers with the same sum of prime factors. more
The first 600 Ruth-Aaron numbers :
5,
6,
8,
9,
15,
16,
24,
25,
49,
50,
77,
78,
104,
105,
125,
126,
153,
154,
369,
370,
492,
493,
714,
715,
948,
949,
1330,
1331,
1520,
1521,
1682,
1683,
1862,
1863,
2107,
2108,
2299,
2300,
2491,
2492,
2600,
2601,
2783,
2784,
3248,
3249,
4185,
4186,
4191,
4192,
5405,
5406,
5560,
5561,
5959,
5960,
6556,
6557,
6811,
6812,
6867,
6868,
8280,
8281,
8463,
8464,
8855,
8856,
9800,
9801,
10647,
10648,
12351,
12352,
12726,
12727,
13775,
13776,
14587,
14588,
16932,
16933,
17080,
17081,
18490,
18491,
18655,
18656,
20450,
20451,
21183,
21184,
24024,
24025,
24432,
24433,
24880,
24881,
24895,
24896,
25839,
25840,
26642,
26643,
26649,
26650,
28448,
28449,
28809,
28810,
33019,
33020,
35456,
35457,
37828,
37829,
37881,
37882,
40081,
40082,
41261,
41262,
42624,
42625,
43215,
43216,
43680,
43681,
44831,
44832,
44891,
44892,
47544,
47545,
48203,
48204,
48762,
48763,
49240,
49241,
52554,
52555,
53192,
53193,
57075,
57076,
61760,
61761,
63344,
63345,
63426,
63427,
63665,
63666,
64232,
64233,
68264,
68265,
68949,
68950,
70356,
70357,
72500,
72501,
75140,
75141,
79118,
79119,
81175,
81176,
89979,
89980,
95709,
95710,
98119,
98120,
98644,
98645,
99163,
99164,
106799,
106800,
106893,
106894,
109939,
109940,
110607,
110608,
112079,
112080,
113283,
113284,
118656,
118657,
128774,
128775,
132209,
132210,
141340,
141341,
141488,
141489,
143622,
143623,
148215,
148216,
154842,
154843,
155111,
155112,
155708,
155709,
172773,
172774,
173162,
173163,
174096,
174097,
180359,
180360,
183446,
183447,
189080,
189081,
197657,
197658,
204258,
204259,
204610,
204611,
208581,
208582,
215391,
215392,
216293,
216294,
224104,
224105,
224256,
224257,
228599,
228600,
230124,
230125,
231284,
231285,
235586,
235587,
244035,
244036,
244595,
244596,
244853,
244854,
247455,
247456,
248109,
248110,
250995,
250996,
261019,
261020,
261568,
261569,
263327,
263328,
263628,
263629,
265112,
265113,
268191,
268192,
268656,
268657,
268715,
268716,
278277,
278278,
289316,
289317,
292454,
292455,
294335,
294336,
296111,
296112,
296751,
296752,
304325,
304326,
319428,
319429,
329988,
329989,
332994,
332995,
333801,
333802,
346692,
346693,
350599,
350600,
351075,
351076,
361063,
361064,
370348,
370349,
373703,
373704,
380438,
380439,
380963,
380964,
381728,
381729,
384312,
384313,
388450,
388451,
394855,
394856,
404973,
404974,
407835,
407836,
416119,
416120,
417162,
417163,
417164,
421819,
421820,
430604,
430605,
436932,
436933,
437715,
437716,
441211,
441212,
445305,
445306,
453474,
453475,
457899,
457900,
461824,
461825,
466773,
466774,
467302,
467303,
470119,
470120,
477092,
477093,
488976,
488977,
493922,
493923,
501830,
501831,
508563,
508564,
509851,
509852,
525224,
525225,
529194,
529195,
531990,
531991,
537916,
537917,
540280,
540281,
547525,
547526,
550032,
550033,
554682,
554683,
557535,
557536,
560150,
560151,
563390,
563391,
564924,
564925,
565823,
565824,
571039,
571040,
572015,
572016,
576984,
576985,
583462,
583463,
587264,
587265,
598249,
598250,
608316,
608317,
618751,
618752,
624079,
624080,
639847,
639848,
643467,
643468,
665322,
665323,
667628,
667629,
671019,
671020,
685228,
685229,
693610,
693611,
694430,
694431,
696436,
696437,
732088,
732089,
732199,
732200,
740726,
740727,
742508,
742509,
742853,
742854,
748230,
748231,
767832,
767833,
773312,
773313,
773604,
773605,
788670,
788671,
801295,
801296,
820600,
820601,
821271,
821272,
823150,
823151,
827895,
827896,
838935,
838936,
839199,
839200,
839375,
839376,
842127,
842128,
844864,
844865,
849375,
849376,
850746,
850747,
851709,
851710,
856607,
856608,
859408,
859409,
861399,
861400,
869054,
869055,
878898,
878899,
882926,
882927,
890126,
890127,
890723,
890724,
901747,
901748,
906019,
906020,
909624,
909625,
911075,
911076,
911537,
911538,
917863,
917864,
927774,
927775,
932034,
932035,
942632,
942633,
944108,
944109,
956039,
956040,
959581,
959582,
961572,
961573,
962140,
962141,
965095,
965096,
976375,
976376,
977280,
977281,
977778,
977779,
977895,
977896,
996557,
996558,
999072,
999073,
1003513,
1003514,
1006549,
1006550,
1019425,
1019426,
1023640,
1023641,
1025381,
1025382,
1039887,
1039888,
1040402,
1040403,
1045950,
1045951,
1053598,
1053599,
1054828,
1054829,
1062874,
1062875,
1065843,
1065844,
1084287,
1084288,
1085875,
1085876,
1095315,
1095316,
1100280,
1100281,
1104263,
1104264,
1131811,
1131812,
1166067,
1166068,
1172947,
1172948,
1173148,
1173149,
1173504,
1173505,
1187144,
1187145,
1191132,
1191133,
1191451,
1191452,
1193983,
1193984,
1199987,
1199988,
1205547,
1205548,
1211898,
1211899,
1232746,
1232747,
1236599,
1236600,
1240965,
1240966,
1243500,
1243501,
1246932.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 1591694 values, from 5 to 999995825316).
n\r | 0 | 1 |
2 | 795847 | 795847 | 2 |
3 | 588958 | 502069 | 500667 | 3 |
4 | 508655 | 395795 | 287192 | 400052 | 4 |
5 | 397964 | 331252 | 265043 | 265001 | 332434 | 5 |
6 | 290763 | 248839 | 251854 | 298195 | 253230 | 248813 | 6 |
7 | 305172 | 252047 | 196201 | 195587 | 196232 | 196729 | 249726 | 7 |
8 | 284022 | 214221 | 144816 | 186025 | 224633 | 181574 | 142376 | 214027 | 8 |
9 | 261247 | 199524 | 150177 | 163529 | 151181 | 150794 | 164182 | 151364 | 199696 | 9 |
10 | 195244 | 162677 | 133266 | 132491 | 166252 | 202720 | 168575 | 131777 | 132510 | 166182 | 10 |
11 | 214166 | 171874 | 129720 | 129193 | 129618 | 129207 | 128771 | 129089 | 128914 | 128891 | 172251 |
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.