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Ruth-Aaron pairs
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 
2795847795847 2 
3588958502069500667 3 
4508655395795287192400052 4 
5397964331252265043265001332434 5 
6290763248839251854298195253230248813 6 
7305172252047196201195587196232196729249726 7 
8284022214221144816186025224633181574142376214027 8 
9261247199524150177163529151181150794164182151364199696 9 
10195244162677133266132491166252202720168575131777132510166182 10 
11214166171874129720129193129618129207128771129089128914128891172251

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.