Total count d_{i,k} of numbers n with lambda(n) = i and s(n) = k and the mean value of s(n) with fixed lambda(n) i |k 0 1 2 3 4 5 6 7 8 9 | mean ---+-------------------------------------------------------------------------------------------+----------- 0 | 1 0 0 0 0 0 0 0 0 0 | 0 1 | 1 1 0 0 0 0 0 0 0 0 | 0.5 2 | 1 2 1 0 0 0 0 0 0 0 | 1 3 | 1 3 4 0 0 0 0 0 0 0 | 1.375 4 | 1 4 9 2 0 0 0 0 0 0 | 1.75 5 | 1 5 17 9 0 0 0 0 0 0 | 2.0625 6 | 1 6 27 29 1 0 0 0 0 0 | 2.359375 7 | 1 7 40 71 9 0 0 0 0 0 | 2.625 8 | 1 8 54 156 37 0 0 0 0 0 | 2.859375 9 | 1 9 70 295 137 0 0 0 0 0 | 3.089844 10 | 1 10 88 514 408 3 0 0 0 0 | 3.295898 11 | 1 11 108 830 1059 39 0 0 0 0 | 3.490234 12 | 1 12 130 1250 2474 229 0 0 0 0 | 3.677490 13 | 1 13 154 1781 5277 966 0 0 0 0 | 3.857666 14 | 1 14 180 2439 10422 3328 0 0 0 0 | 4.029480 15 | 1 15 208 3222 19375 9945 2 0 0 0 | 4.191101 16 | 1 16 238 4136 34082 26952 111 0 0 0 | 4.343475 17 | 1 17 270 5193 56653 67725 1213 0 0 0 | 4.491051 18 | 1 18 304 6403 89423 158275 7720 0 0 0 | 4.635704 19 | 1 19 340 7773 134226 346078 35851 0 0 0 | 4.780613 20 | 1 20 378 9320 192950 712196 133711 0 0 0 | 4.924566 21 | 1 21 418 11043 266972 1389147 429549 1 0 0 | 5.066352 22 | 1 22 460 12960 358195 2585664 1236898 104 0 0 | 5.203018 23 | 1 23 504 15075 467785 4610539 3290778 3903 0 0 | 5.333672 24 | 1 24 550 17406 597711 7883337 8221683 56504 0 0 | 5.458981 25 | 1 25 598 19949 749580 12909574 19428175 446530 0 0 | 5.582035 26 | 1 26 648 22730 925801 20244557 43506741 2408360 0 0 | 5.705572 27 | 1 27 700 25744 1127821 30417906 92565735 10079794 0 0 | 5.831066 28 | 1 28 754 29012 1358492 43957633 187703987 35385549 0 0 | 5.957609 29 | 1 29 810 32533 1619320 61331573 364375132 109511480 34 0 | 6.083522 30 | 1 30 868 36330 1913410 83025830 680066847 308690716 7792 0 | 6.206512 31 | 1 31 928 40396 2242427 109512859 1223977352 811352999 356655 0 | 6.325005 32 | 1 32 990 44761 2609901 141319527 2125933798 2018866416 6191870 0 | 6.438786 33 | 1 33 1054 49411 3017288 178951087 3558123266 4792148588 57643864 0 | 6.549748 34 | 1 34 1120 54378 3468569 223025952 5727896722 10870732950 354689458 0 | 6.660656 35 | 1 35 1188 59654 3965281 274130723 8859290834 23576066805 1646223847 0 | 6.773762 36 | 1 36 1258 65267 4511668 332958287 13179662011 48940977928 6261300280 0 | 6.889433 37 | 1 37 1330 71202 5109307 400185236 18906905691 97493267156 20633413500 12 | 7.006625 38 | 1 38 1404 77496 5762769 476612449 26254767265 | 39 | 1 39 1480 84134 6473591 562976022 35429206175 | 40 | 1 40 1558 91149 7246968 660190451 | 41 | 1 41 1638 98526 8084084 769070413 | 42 | 1 42 1720 106301 8990542 890636193 | 43 | 1 43 1804 114454 9967529 1025795022 | 44 | 1 44 1890 123029 11021308 1175677920 | 45 | 1 45 1978 132000 12152774 1341256012 | 46 | 1 46 2068 141409 13368578 1523806404 | 47 | 1 47 2160 151237 14669675 1724387403 | 48 | 1 48 2254 161523 16062933 1944373820 | 49 | 1 49 2350 172242 17549235 2184924626 | 50 | 1 50 2448 183443 19136233 2447563221 | 51 | 1 51 2548 195097 20824497 2733519725 | 52 | 1 52 2650 207250 22622202 3044485481 | 53 | 1 53 2754 219874 24529454 3381755036 | 54 | 1 54 2860 233019 26555193 3747159160 | 55 | 1 55 2968 246653 28699540 4142110371 | 56 | 1 56 3078 260830 30971914 4568600823 | 57 | 1 57 3190 275513 33371951 5028107937 | 58 | 1 58 3304 290755 35909500 5522804082 | 59 | 1 59 3420 306527 38584411 6054263301 | 60 | 1 60 3538 322880 41407278 | 61 | 1 61 3658 339775 | 62 | 1 62 3780 357274 | 63 | 1 63 3904 375335 | Trivially holds: k > i ==> d_{i,k} = 0 d_{i,k}=0 ==> \forall j >= k: d_{i,j} = 0 \sum_{k=0}^i d_{i,k} = 2^i By definition: \sum_{k=0}^r d_{r-k,k} = d(r) mean_i = 2^-i * \sum_{k=0}^i k*d_{i,k} Conjectures: d_{i,k}=0 ==> \forall h <= i: d_{h,k} = 0 d_{i+1,k} >= d_{i,k} \forall k \exists C_k > 0: \lim_{i -> \infty} d_{i,k} / i^k = C_k Last updated: 2020-05-03 21:45 UTC+2