algorithm - 用给定数量的因子找到最小数字的算法

Question

任何人都能想到的最有效的算法是什么，给定一个自然数n，返回具有n 个正除数（包括 1 和x ）的最小自然数x？例如，给定 4，算法应该得到 6（除数：1,2,3,6）；即 6 是具有 4 个不同因子的最小数字。同样，给定 6，算法应该得到 12（除数：1、2、3、4、6、12）；即 12 是具有 6 个不同因子的最小数字

就实际性能而言，我正在寻找一种可扩展的算法，它可以在每秒可以进行 10 ⁷次计算的机器上在 2 秒内给出 10 ^{20量级的答案。}

Answer 1

http://www.primepuzzles.net/problems/prob_019.htm

b) Jud McCranie、TWA Baumann 和 Enoch Haga 发送基本相同的程序来找到给定d 的N(d) ：

1. 将d分解为他的主要因数的乘积：d = p ₁ ^{a ₁} * p ₂ ^{a ₂} *p ₃ ^{a ₃} *...
2. 将此分解转换为另一个算术等效分解，由非幂单调递减且不一定是素因数组成...（uf！...）d = p ₁ ^{a ₁} * p ₂ ^{a ₂} *p ₃ ^{a ₃} *.. . = b ₁ * b ₂ * b ₃ ...这样b ₁ ≥ b ₂ ≥ b ₃ ...
   您必须意识到，对于每个给定d的 ，可以进行几个算术等价的因式分解：例如：
   如果d = 16 = 2 ⁴那么有 5 个等价分解： d = 2*2*2*2 = 4*2*2 = 4*4 = 8*2 = 16
3. N是对 d 的所有等价分解计算2 ^{b ₁ -1} * 3 ^{b ₂ -1} * 5 ^{b ₃ -1} * ...的最小结果。使用相同的示例：
   N(16) = 这些 {2 * 3 * 5 * 7, 2 ³ * 3 * 5, 2 ³ * 3 ³ , 2 ⁷ * 3, 2 ¹⁵ } = 2 ³ * 3 * 5 = 120

更新：数字在 10 ²⁰左右，请注意同一页上引用的 Christian Bau 的注释。

Answer 2

//What is the smallest number with X factors?
function smallestNumberWithThisManyFactors(factorCount) {

  Number.prototype.isPrime = function() {
    let primeCandidate = this;
    if(primeCandidate <= 1 || primeCandidate % 1 !== 0) return false
    let i = 2;
    while(i <= Math.floor(Math.sqrt(primeCandidate))){
      if(primeCandidate%i === 0) return false;
      i++;
    }
    return true;
  }

  Number.prototype.nextPrime = function() {
    let currentPrime = this;
    let nextPrimeCandidate = currentPrime + 1
    while(nextPrimeCandidate < Infinity) {
      if(nextPrimeCandidate.isPrime()){
        return nextPrimeCandidate;
      } else {
        nextPrimeCandidate++;
      }
    }
  }

  Number.prototype.primeFactors = function() {
    let factorParent = this;
    let primeFactors = [];
    let primeFactorCandidate = 2;
    while(factorParent !== 1){
      while(factorParent % primeFactorCandidate === 0){
        primeFactors.push(primeFactorCandidate);
        factorParent /= primeFactorCandidate;
      }
      primeFactorCandidate = primeFactorCandidate.nextPrime();
    }
    return primeFactors;
  }

  Number.prototype.factors = function() {
    let parentNumber = this.valueOf();
    let factors = []
    let iterator = parentNumber % 2 === 0 ? 1 : 2

    let factorCandidate = 1;
    for(factorCandidate; factorCandidate <= Math.floor(parentNumber/2); factorCandidate += iterator) {
      if(parentNumber % factorCandidate === 0) {
        factors.push(factorCandidate)
      }
    }
    factors.push(parentNumber)
    return factors
  }

  Array.prototype.valueSort = function() {
    return this.sort(function (a,b){ return a-b })
  }

  function clone3DArray(arrayOfArrays) {
    let cloneArray = arrayOfArrays.map(function(arr) {
      return arr.slice();
    });
    return cloneArray;
  }

  function does3DArrayContainArray(arrayOfArrays, array){
    let aOA = clone3DArray(arrayOfArrays);
    let a = array.slice(0);
    for(let i=0; i<aOA.length; i++){
      if(aOA[i].sort().join(',') === a.sort().join(',')){
        return true;
      }
    }
    return false;
  }

  function removeDuplicateArrays(combinations) {
    let uniqueCombinations = []
    for(let c = 0; c < combinations.length; c++){
      if(!does3DArrayContainArray(uniqueCombinations, combinations[c])){
        uniqueCombinations[uniqueCombinations.length] = combinations[c];
      }
    }
    return uniqueCombinations;
  }

  function generateCombinations(parentArray) {
    let generate = function(n, src, got, combinations) {
      if(n === 0){
        if(got.length > 0){
          combinations[combinations.length] = got;
        }
        return;
      }
      for (let j=0; j<src.length; j++){
        generate(n - 1, src.slice(j + 1), got.concat([src[j]]), combinations);
      }
      return;
    }
    let combinations = [];
    for(let i=1; i<parentArray.length; i++){
      generate(i, parentArray, [], combinations);
    }
    combinations.push(parentArray);
    return combinations;
  }

  function generateCombinedFactorCombinations(primeFactors, primeFactorCombinations) {
    let candidates = [];
    for(let p=0; p<primeFactorCombinations.length; p++){
      let product = 1;
      let primeFactorsCopy = primeFactors.slice(0);
      for(let q=0; q<primeFactorCombinations[p].length; q++){
        product *= primeFactorCombinations[p][q];
        primeFactorsCopy.splice(primeFactorsCopy.indexOf(primeFactorCombinations[p][q]), 1);
      }
      primeFactorsCopy.push(product);
      candidates[candidates.length] = primeFactorsCopy.valueSort().reverse();
    }
    return candidates;
  }

  function determineMinimumCobination (candidates){
    let minimumValue = Infinity;
    let bestFactorCadidate = []
    for(let y=0; y<candidates.length; y++){
      let currentValue = 1;
      let currentPrime = 2;
      for(let z=0; z<combinedFactorCandidates[y].length; z++){
        currentValue *= Math.pow(currentPrime,(combinedFactorCandidates[y][z])-1);
        currentPrime = currentPrime.nextPrime();
      }
      if(currentValue < minimumValue){
        minimumValue = currentValue;
        bestFactorCadidate = combinedFactorCandidates[y];
      }
    }
    return minimumValue;
  }

  let primeFactors = factorCount.primeFactors();
  let primeFactorCombinations = removeDuplicateArrays(generateCombinations(primeFactors));
  let combinedFactorCandidates = generateCombinedFactorCombinations(primeFactors, primeFactorCombinations);
  let smallestNumberWithFactorCount = determineMinimumCobination(combinedFactorCandidates);
  console.log('The smallest number with ' + factorCount + ' factors is: ')
  console.log(smallestNumberWithFactorCount)
  console.log('With these factors being: ')
  console.log(smallestNumberWithFactorCount.factors())

  return smallestNumberWithFactorCount;
}

smallestNumberWithThisManyFactors(10)