add some OEIS links

Mallji · Jun 18, 2020 · 80134db · 80134db
1 parent 257b224
commit 80134db
Show file tree

Hide file tree

Showing 3 changed files with 48 additions and 28 deletions.
diff --git a/copypasta/common.go b/copypasta/common.go
@@ -40,7 +40,6 @@ import (
 func commonCollection() {
 	const alphabet = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
 	pow10 := [...]int{1, 1e1, 1e2, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8, 1e9} // math.Pow10
-	factorial := [...]int{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 /*10!*/, 39916800, 479001600}
 	// TIPS: dir4[i] 和 dir4[i^1] 互为相反方向
 	type pair struct{ x, y int }
 	dir4 := [...]pair{{-1, 0}, {1, 0}, {0, -1}, {0, 1}} // 上下左右

diff --git a/copypasta/math.go b/copypasta/math.go
@@ -650,11 +650,12 @@ func numberTheoryCollection() {
 	}
 
 	// 欧拉函数（互质的数的个数）Euler totient function https://oeis.org/A000010
-	// 预处理所有 [2,mx] 数的欧拉函数
-	// NOTE: phi 的迭代（指 phi[phi...[n]]）是 log 级别收敛的：奇数减一，偶数减半
-	phiAll := func() {
+	// https://en.wikipedia.org/wiki/Euler%27s_totient_function
+	// 预处理 [1,mx] 欧拉函数
+	// NOTE: phi[phi...[n]] 收敛到 1 的迭代次数是 log 级别的：奇数减一，偶数减半 https://oeis.org/A003434
+	initPhi := func() {
 		const mx int = 1e6
-		phi := [mx + 1]int{}
+		phi := [mx + 1]int{1: 1}
 		for i := 2; i <= mx; i++ {
 			phi[i] = i
 		}
@@ -680,7 +681,7 @@ func numberTheoryCollection() {
 
 	// Unitary totient (or unitary phi) function uphi(n) http://oeis.org/A047994
 
-	/* 同余 */
+	/* 同余 逆元 */
 
 	// 二元一次不定方程
 	// exgcd solve equation ax+by=gcd(a,b)
@@ -838,11 +839,12 @@ func numberTheoryCollection() {
 	// todo
 	// 模板题 https://www.luogu.com.cn/problem/P5491 https://www.luogu.com.cn/problem/P5668
 
-	/* 组合数；二项式系数
-	 */
+	/* 阶乘 组合数/二项式系数 */
 
-	// 阶乘
-	factorial := func(n int) int64 {
+	// https://oeis.org/A000142
+	factorial := [...]int{1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800 /*10!*/, 39916800, 479001600}
+
+	calcFactorial := func(n int) int64 {
 		res := int64(1) % mod
 		for i := 2; i <= n; i++ {
 			res = res * int64(i) % mod
@@ -1060,14 +1062,25 @@ func numberTheoryCollection() {
 	// 浅谈一类积性函数的前缀和 + 套题 https://blog.csdn.net/skywalkert/article/details/50500009
 	// 模板题 https://www.luogu.com.cn/problem/P4213
 
+	// 埃及分数 - 不同的单位分数的和 (IDA*)
+	// https://www.luogu.com.cn/problem/UVA12558
+	// 贪婪算法：将一项分数分解成若干项单分子分数后的项数最少，称为第一种好算法；最大的分母数值最小，称为第二种好算法
+	// https://en.wikipedia.org/wiki/Egyptian_fraction
+	// https://oeis.org/A006585 number of solutions
+	// https://oeis.org/A247765 Table of denominators in the Egyptian fraction representation of n/(n+1) by the greedy algorithm
+	// https://oeis.org/A100678 Number of Egyptian fractions in the representation of n/(n+1) via the greedy algorithm
+	// https://oeis.org/A100695	Largest denominator used in the Egyptian fraction representation of n/(n+1) by the greedy algorithm
+	//
+	// 		埃尔德什-施特劳斯猜想（Erdős–Straus conjecture）https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Straus_conjecture
+
 	_ = []interface{}{
 		primes,
 		sqCheck, cubeCheck, sqrt, cbrt, bottomDiff,
 		gcd, gcdPrefix, gcdSuffix, lcm, frac, cntRangeGCD,
 		isPrime, sieve, primeFactorization, primeDivisors, primeExponentsCountAll,
-		divisors, divisorPairs, doDivisors, doDivisors2, oddDivisorsNum, maxSqrtDivisor, divisorsAll, primeFactorsAll, lpfAll, distinctPrimesCountAll, calcPhi, phiAll,
+		divisors, divisorPairs, doDivisors, doDivisors2, oddDivisorsNum, maxSqrtDivisor, divisorsAll, primeFactorsAll, lpfAll, distinctPrimesCountAll, calcPhi, initPhi,
 		exgcd, invM, invP, divM, divP, initAllInv, crt, excrt, babyStepGiantStep,
-		factorial, initFactorial, _factorial, combHalf, comb,
+		factorial, calcFactorial, initFactorial, _factorial, combHalf, comb,
 		muInit,
 	}
 }
@@ -1111,6 +1124,12 @@ Stern-Brocot 树与 Farey 序列 https://oi-wiki.org/misc/stern-brocot/ https://
 https://en.wikipedia.org/wiki/Generating_function
 整数分拆 https://oeis.org/A000041 https://en.wikipedia.org/wiki/Partition_(number_theory)
 
+	质数分拆
+	https://oeis.org/A061358 Number of ways of writing n=p+q with p, q primes and p>=q
+	https://oeis.org/A067187 Numbers that can be expressed as the sum of two primes in exactly one way
+	https://oeis.org/A068307 number of partitions of n into a sum of three primes
+	https://oeis.org/A071335 Number of partitions of n into a sum of at most three primes
+
 记 A = [1,2,...,n]，A 的全排列中与 A 的最大差值为 n^2/2 https://oeis.org/A007590
 Maximum sum of displacements of elements in a permutation of (1..n)
 For example, with n = 9, permutation (5,6,7,8,9,1,2,3,4) has displacements (4,4,4,4,4,5,5,5,5) with maximal sum = 40

diff --git a/copypasta/math_continued_fraction.go b/copypasta/math_continued_fraction.go
@@ -28,8 +28,10 @@ func continuedFractionCollections() {
 	}
 
 	// sqrt(d) = [exp[0]; exp[1],..., 2*exp[0], exp[1], ..., 2*exp[0], exp[1], ...]
-	// https://en.wikipedia.org/wiki/Pell%27s_equation
+	// https://en.wikipedia.org/wiki/Pell%27s_equation 解 https://oeis.org/A002350 https://oeis.org/A002349
 	// https://www.weiwen.io/post/about-the-pell-equations-2/
+	// 连分数表示 https://oeis.org/A240071
+	// 循环节长度 https://oeis.org/A003285
 	calcSqrtContinuedFraction := func(d int64) (exp []int64) {
 		sqrtD := math.Sqrt(float64(d))
 		base := int64(sqrtD)
@@ -77,16 +79,16 @@ func continuedFractionCollections() {
 	//	return
 	//}
 
-	calcGCD := func(a, b int64) int64 {
+	gcd := func(a, b int64) int64 {
 		for b > 0 {
 			a, b = b, a%b
 		}
 		return a
 	}
-	calcGCDN := func(nums ...int64) (gcd int64) {
-		gcd = nums[0]
-		for _, v := range nums[1:] {
-			gcd = calcGCD(gcd, v)
+	gcds := func(a ...int64) (g int64) {
+		g = a[0]
+		for _, v := range a[1:] {
+			g = gcd(g, v)
 		}
 		return
 	}
@@ -111,11 +113,11 @@ func continuedFractionCollections() {
 			if newC == 0 {
 				return
 			}
-			gcd := calcGCDN(newA, newB, newC)
-			//Println(i, base, newA, newB, newC, gcd)
-			a = append(a, newA/gcd)
-			b = append(b, newB/gcd)
-			c = append(c, newC/gcd)
+			g := gcds(newA, newB, newC)
+			//Println(i, base, newA, newB, newC, g)
+			a = append(a, newA/g)
+			b = append(b, newB/g)
+			c = append(c, newC/g)
 		}
 		return
 	}
@@ -147,14 +149,14 @@ func continuedFractionCollections() {
 			if newC.IsInt64() && newC.Int64() == 0 {
 				return
 			}
-			gcd := big.NewInt(1)
+			g := big.NewInt(1)
 			if !base.IsInt64() || base.Int64() > 0 {
-				gcd.GCD(nil, nil, newA, newB).GCD(nil, nil, gcd, newC)
+				g.GCD(nil, nil, newA, newB).GCD(nil, nil, g, newC)
 			}
 			//Println(i, base, newA, newB, newC, gcd)
-			a = append(a, newA.Quo(newA, gcd))
-			b = append(b, newB.Quo(newB, gcd))
-			c = append(c, newC.Quo(newC, gcd))
+			a = append(a, newA.Quo(newA, g))
+			b = append(b, newB.Quo(newB, g))
+			c = append(c, newC.Quo(newC, g))
 		}
 		return
 	}