Numbers
Book of Numbers, 圣经的第四书 但这篇记录一些有趣的数组【不是刻意去认识那些的】
1. Perfect Number Definition: 6 = 1 x 2 x 3 6 = 1 + 2 + 3 [1, 2, 3] Divisors of 6. List: 1. 6, 2. 28, 3. 496, 4. 8128, 5. 33550336, 6. 8589869056, 7. 137438691328, 8. 2305843008139952128, 9. 2658455991569831744654692615953842176, 10. 191561942608236107294793378084303638130997321548169216 ... Algorithm: Euclid=Euler Theorem, for even perfect numbers n = 2^(p - 1) * (2^p - 1)
2. Fibonacci Sequence Definition: 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 ... List: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025...] Relative terms: Phi Appro. = f(n+1)/f(n)
3. Palindromic Prime Definition: When a Prime is symmetric, it is a PP... List: [2,3,5,7,11,101,131,151,181,191,313,353,373,383,727,757,787,797,919,929,10301,10501,10601, 11311,11411,12421,12721,12821,13331,13831,13931,14341,14741,15451,15551,16061,16361, 16561,16661,17471, 17971,18181...]
4. Circular Prime Definition: When rotate a prime, each number you get is a prime, then it is a CP... List: [2,3,5,7,11,13,17,37,79,113,197,199,337,1193,3779,11939,19937,193939,199933, 1111111111111111111, 11111111111111111111111] Algorithm:
终于明白为什么以前看到数论后不久就远远避开了 这是有多无趣 而且还跨越了进制...
5. Truncatable Prime Definition: When eliminate a number in a prime from left/right, the remmaining number is still a prime, then it is a TP... List: [both left and right] [2,3,5,7,23,37,53,73,313,317,373,797,3137,3797, 739397] All numbers are in the list...
6. Imaginary Numbers
7. Bellman-Ford Algorithm Finding the shortest path in a graph or matrix, going through all the paths... O(|V| * |E|), |V| and |E| are the number of vertices and edges.
function BellmanFord(list vertices, list edges, vertex source) ::distance[],predecessor[] // This implementation takes in a graph, represented as // lists of vertices and edges, and fills two arrays // (distance and predecessor) with shortest-path // (less cost/distance/metric) information // Step 1: initialize graph for each vertex v in vertices: distance[v] := inf // At the beginning , all vertices have a weight of infinity predecessor[v] := null // And a null predecessor distance[source] := 0 // The weight is zero at the source // Step 2: relax edges repeatedly for i from 1 to size(vertices)-1: for each edge (u, v) with weight w in edges: if distance[u] + w < distance[v]: distance[v] := distance[u] + w predecessor[v] := u // Step 3: check for negative-weight cycles for each edge (u, v) with weight w in edges: if distance[u] + w < distance[v]: error "Graph contains a negative-weight cycle" return distance[], predecessor[] Related: Dijkstra's algorithm, Floyd-Warshall algorithm I really don't wanna learn these things...
8. Euler Totient Function
9. Ramanujan Summation 1 + 2 + 3 + 4 + 5 + ... = -1/12 While using Riemann Zeta function, you will get the same result, as well as Cutoff regularization.
The identity (1-2^{1-s})ζ (s)=η(s) continues to hold when both functions are extended by analytic continuation to include values of s for which the above series diverge. Substituting s = −1, one gets −3ζ(−1) = η(−1). Now, computing η(−1) is an easier task, as the eta function is equal to the Abel sum of its defining series, which is a one-sided limit:
Thus ζ(−1) = − 1/12
10. Pascal Triangle Column[Table[Binomial[n, k], {n, 0, 10}, {k, 0, n}], Center]
References: en.wikipedia.org oeis.org
Words, which I learnt from varies articles Serendipitous/Serendipity An unsought, unintended, and/or unexpected, but fortunate, discovery and/or learning experience that happens by accident. A combination of events which are not individually beneficial, but occurring together to produce a good or wonderful outcome. Corroborate: To confirm or support something with additional evidence; to attest or vouch for. To make strong; to strengthen. Insurmountable Incapable of being passed over Tantamount Equal to
