数字串的通项公式

王守恩

A003348——Numbers that are the sum of 3 positive 5th powers.——A344641是同一个数字串。

3, 34, 65, 96, 245, 276, 307, 487, 518, 729, 1026, 1057, 1088, 1268, 1299, 1510, 2049, 2080, 2291, 3072, 3127, 3158, 3189, 3369, 3400, 3611, 4150, 4181, 4392, 5173, 6251, 6282, 6493, 7274, 7778, 7809, 7840, 8020, 8051, 8262, 8801, 8832,

Union[n /. Solve[{a^5 + b^5 + c^5 == n, 354810000 > n > 354000000, a >= b >= c > 0}, {n, a, b, c}, Integers]]——用我们的公式也可以出来前20000项。

{353509243, 353522298, 353524961, 353525699, 353528450, 353708774, 353730794, 353747250, 353798400, 353864576, 353937693, 353987492, 354010368, 354015599, 354017056, 354019995, 354036451, 354111492, 354132448,
354262984, 354295305, 354354750, 354391276, 354405537, 354407732, 354483062, 354542101, 354550285, 354597981, 354661594, 354790877, 354790908, 354791119, 354791900, 354793792, 354794001, 354798652, 354807683}

19963 353509243
19964 353522298
19965 353524961
19966 353525699
19967 353528450
19968 353708774
19969 353730794
19970 353747250
19971 353798400
19972 353864576
19973 353937693
19974 353987492
19975 354010368
19976 354015599
19977 354017056
19978 354019995
19979 354036451
19980 354111492
19981 354132448
19982 354262984
19983 354295305
19984 354354750
19985 354391276
19986 354405537
19987 354407732
19988 354483062
19989 354542101
19990 354550285
19991 354597981
19992 354661594
19993 354790877
19994 354790908
19995 354791119
19996 354791900
19997 354793792
19998 354794001
19999 354798652
20000 354807683

王守恩

A046080——\(\D n^2 = a_{1}^2 + a_{2}^2\)
0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 1, 0, 1, 0, 0, 1, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 0, 4, 0, 1, 0, 1, 1, 1, 0, 0, 0,

A181786——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2\)
0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 0, 2, 1, 1, 1, 3, 0, 2, 3, 3, 0, 6, 2, 3, 1, 2, 1, 8, 1, 3, 3, 4, 0, 10, 2, 5, 3, 4, 3, 8, 0, 5, 6, 6, 2, 11, 3, 6, 1, 8, 2, 12, 1, 6, 8, 8, 1, 15, 3, 8, 3, 7, 4, 20, 0, 6, 10, 9, 2, 16, 5, 9, 3, 9, 4, 15, 3, 15, 8, 10, 0, 22, 5, 11, 6, 9, 6, 18,

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2\)
{0, 1, 0, 1, 1, 2, 2, 1, 2, 5, 3, 2, 5, 8, 9, 1, 7, 10, 9, 5, 16, 12, 13, 2, 19, 18, 22, 8, 20, 32, 23, 1, 35, 25, 42, 10, 32, 31, 51, 5, 38, 55, 42, 12, 80, 43, 50, 2, 63, 62, 83, 18, 63, 75, 91, 8, 103, 65, 77, 32, 83, 74, 144, 1, 127, 116, 99, 25, 151, 133}

A179015——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2\)
0, 0, 0, 1, 1, 1, 2, 5, 2, 6, 6, 9, 9, 15, 8, 25, 20, 21, 25, 39, 26, 46, 44, 57, 49, 71, 52,102, 81, 81, 99, 145,92, 156, 126,164, 160, 204, 151, 247, 217, 236, 245, 326, 211, 357, 319, 381, 360, 416, 344, 518, 446, 476, 450, 670, 468, 675, 607, 661,

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2\)
{0, 0, 1, 0, 1, 4, 3, 2, 10, 10, 12, 16, 20, 28, 40, 32, 46, 81, 68, 84, 107, 138, 131, 160, 184, 250, 253, 272, 296, 459, 374, 452, 501, 650, 599, 732, 714,981, 901, 1124, 1036, 1405, 1236, 1566, 1558, 1994, 1718, 2086, 1961, 2873, 2324, 3004}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2 +a_{7}^2 \)
{0, 0, 0, 1, 2, 1, 4, 7, 9, 13, 19, 29, 35, 46, 68, 87, 101, 138, 159, 212, 261, 308, 348, 505, 488, 633, 756, 905, 936, 1268, 1250, 1647, 1808, 2113, 2078, 3022, 2726, 3508, 3755, 4400, 4320, 5956, 5357, 6984, 7025, 8427, 8015, 11241, 9659}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2+a_{7}^2  +a_{8}^2 \)
{0, 0, 0, 1, 1, 1, 5, 9, 9, 15, 30, 32, 61, 70, 93, 149, 202, 198, 338, 421, 486, 608, 845, 942, 1280, 1423, 1719, 2246,  2673, 2838, 3752,  4478, 4869, 5736, 7094, 7912, 9356, 10363, 11844, 14715, 16045, 17316, 20649, 24188, 25767, 29132}

OEIS没有——\(\D n^2 = a_{1}^2 + a_{2}^2 +a_{3}^2 +a_{4}^2 +a_{5}^2 +a_{6}^2 +a_{7}^2 +a_{8}^2 +a_{9}^2 \)
{0, 0, 1, 0, 1, 4, 3, 5, 16, 19, 27, 52, 63, 95, 155, 182, 260, 389, 479, 626, 880, 1088,1401, 1792, 2265, 2811, 3599, 4275,5386, 6507, 7998,9398, 11620, 13690, 16545, 19027, 23217, 26841, 31677, 36210, 43529, 48992, 58411, 65608, 75993}

可以有统一的通项公式——Count[PowersRepresentations[#^2, 9, 2], w_ /; (Times@@w) > 0] & /@Range[50]