This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
From _Daniel Forgues_, Aug 07 2018: (Start)
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
From _Gus Wiseman_, May 19 2020: (Start)
The sequence together with the corresponding partitions begins:
1: () 24: (2,1,1,1) 35: (4,3)
2: (1) 32: (1,1,1,1,1) 42: (4,2,1)
3: (2) 13: (6) 56: (4,1,1,1)
4: (1,1) 22: (5,1) 50: (3,3,1)
5: (3) 21: (4,2) 45: (3,2,2)
6: (2,1) 28: (4,1,1) 60: (3,2,1,1)
8: (1,1,1) 25: (3,3) 80: (3,1,1,1,1)
7: (4) 30: (3,2,1) 54: (2,2,2,1)
10: (3,1) 40: (3,1,1,1) 72: (2,2,1,1,1)
9: (2,2) 27: (2,2,2) 96: (2,1,1,1,1,1)
12: (2,1,1) 36: (2,2,1,1) 128: (1,1,1,1,1,1,1)
16: (1,1,1,1) 48: (2,1,1,1,1) 19: (8)
11: (5) 64: (1,1,1,1,1,1) 34: (7,1)
14: (4,1) 17: (7) 39: (6,2)
15: (3,2) 26: (6,1) 52: (6,1,1)
20: (3,1,1) 33: (5,2) 55: (5,3)
18: (2,2,1) 44: (5,1,1) 66: (5,2,1)
(End)
b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10); # Alois P. Heinz, Feb 14 2020
Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *) b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]]; T[n_] := b[n, n]; T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)
a(1)=77 because the power tree construction produces the chain 1 2 3 5 7 14 19 38 76 77 requiring 9 additions, whereas there are 4 shortest chains that come along with 8 additions, e.g. 1 2 4 8 9 17 34 43 77.
a(1)=8719 because this is the smallest number for which the addition chain produced by the power tree method [1 2 3 5 7 14 28 56 61 117 234 468 936 1872 3744 3861 7722 7783 8719] is by two terms longer than the shortest possible chains for this number. An example of such a chain is [1 2 3 6 9 15 17 34 68 136 272 544 1088 2176 4352 4367 8719].
a(1)=6475341 because this is the smallest number for which the addition chain produced by the power tree method [1 2 3 5 7 14 19 38 76 79 158 316 632 1264 2528 5056 5063 10119 12647 25294 50588 101176 202352 404704 809408 809427 1618835 3237670 6475340 6475341] is by three terms longer than the shortest possible chains for this number. An example of such a chain is [1 2 4 8 16 32 64 65 129 258 387 774 1548 1613 3161 6322 12644 25288 50576 101152 202304 404608 809216 1618432 3236864 3238477 6475341].
The power tree sequence for 54 is 1,2,3,6,9,18,27,54, so a(54) = 27.
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