Displaying 1-10 of 11 results found.
Integer floor of the reciprocal of Gauss-Kuzmin distribution of n: a(n) = floor( log(2)/log(1 + 1/(n*(n+2))) ).
+10
11
2, 5, 10, 16, 24, 33, 44, 55, 68, 83, 99, 116, 135, 155, 177, 199, 224, 249, 276, 305, 335, 366, 398, 432, 468, 504, 543, 582, 623, 665, 709, 754, 800, 848, 897, 948, 1000, 1053, 1108, 1164, 1222, 1281, 1341, 1403, 1466, 1530, 1596, 1663, 1732, 1802, 1873, 1946
PROG
(PARI) for(n=1, 100, a=floor(log(2)/log(1+1/(n*(n+2)))); print1(a, ", "))
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of k in order by n.
+10
3
1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 3, 1, 2, 1, 5, 1, 1, 2, 1, 3, 6, 1, 4, 2, 1, 1, 1, 2, 3, 1, 7, 1, 2, 1, 5, 1, 4, 2, 1, 3, 1, 8, 1, 2, 1, 1, 3, 2, 1, 6, 1, 4, 9, 1, 2, 1, 5, 1, 3, 2, 1, 1, 1, 2, 10, 1, 4, 3, 1, 7, 2, 1, 1, 1, 2, 1, 3, 5, 1, 11, 2, 6, 1, 4, 1, 2, 1, 3, 1, 1, 8, 2, 1, 1, 12, 2, 1, 3, 4, 1, 1
COMMENTS
The geometric mean of the sequence equals Khintchine's constant K=2.685452001 = A002210 since the frequency of the integers agrees with the Gauss-Kuzmin distribution. When considered as a continued fraction, the resulting constant is 0.5815803358828329856145... = A372869 = [0;1,1,2,1,1,3,2,1,1,1,4,2,1,...].
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of n when k=2.
+10
3
3, 7, 12, 16, 21, 27, 31, 36, 41, 47, 51, 58, 63, 67, 74, 78, 84, 89, 95, 99, 106, 110, 116, 123, 127, 133, 137, 142, 149, 156, 160, 166, 170, 177, 183, 187, 192, 199, 205, 210, 216, 221, 227, 232, 236, 243, 249, 256, 259, 265, 270, 278, 284, 287, 294, 297, 305
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of floor(y) in order by n.
+10
2
2, 4, 5, 7, 9, 10, 11, 12, 14, 16, 16, 17, 19, 21, 21, 23, 24, 24, 26, 28, 29, 31, 32, 33, 33, 33, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 50, 52, 53, 53, 55, 55, 57, 58, 60, 62, 64, 64, 65, 67, 67, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 84, 85
COMMENTS
Missing positive integers are found in A084578.
Positive integers not found in A084577.
+10
2
1, 3, 6, 8, 13, 15, 18, 20, 22, 25, 27, 30, 34, 37, 39, 46, 51, 54, 56, 59, 61, 63, 66, 71, 78, 80, 87, 90, 92, 95, 97, 102, 104, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 140, 143, 145, 148, 157, 160, 162, 165, 174, 179, 181, 184, 189, 191, 208, 210, 213
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of m in order by n.
+10
2
1, 2, 1, 3, 4, 1, 2, 5, 6, 7, 1, 3, 8, 2, 9, 4, 10, 1, 11, 12, 5, 13, 3, 1, 14, 2, 6, 15, 16, 17, 7, 4, 18, 1, 19, 8, 20, 2, 21, 3, 9, 22, 5, 23, 1, 24, 10, 25, 26, 6, 11, 27, 2, 28, 4, 1, 29, 12, 30, 3, 31, 7, 13, 32, 33, 34, 14, 1, 35, 5, 8, 36, 2, 15, 37, 38, 39, 16, 40, 9, 4, 41, 1, 17, 3
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of n when k=1.
+10
2
1, 2, 4, 5, 8, 9, 10, 13, 15, 17, 19, 20, 22, 25, 28, 29, 30, 33, 35, 37, 39, 42, 44, 46, 48, 49, 52, 54, 57, 59, 61, 64, 65, 66, 69, 72, 75, 76, 77, 79, 82, 86, 88, 90, 92, 93, 96, 97, 100, 103, 104, 107, 108, 111, 113, 115, 119, 122, 124, 125, 126, 129, 131, 134, 136
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of n when k=4.
+10
2
11, 26, 40, 55, 70, 87, 102, 118, 132, 148, 164, 180, 195, 213, 228, 245, 260, 276, 292, 308, 326, 342, 355, 375, 391, 407, 421, 439, 454, 473, 487, 505, 521, 536, 552, 568, 587, 602, 615, 636, 650, 669, 684, 699, 718, 732, 750, 767, 783, 802, 816, 831, 849
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of n when k=5.
+10
2
18, 38, 60, 81, 105, 128, 152, 172, 196, 219, 242, 266, 289, 313, 337, 358, 385, 408, 431, 453, 479, 500, 525, 549, 574, 597, 617, 644, 671, 691, 716, 738, 762, 789, 811, 834, 858, 884, 908, 929, 956, 977, 1002, 1023, 1052, 1075, 1096
Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of n when k=6.
+10
2
24, 53, 85, 114, 146, 178, 209, 240, 273, 306, 338, 370, 401, 434, 467, 498, 531, 563, 596, 629, 660, 692, 724, 759, 793, 822, 856, 889, 921, 955, 987, 1018, 1056, 1085
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