Введение
Поддающееся число — это положительное целое число, для которого существует мультимножество, состоящее из такого же количества целых чисел, как и само число, которое в сумме даёт исходное число, а при перемножении — тоже даёт исходное число. Если выразить это алгебраически, для положительного целого числа *n* существует мультимножество из *n* целых чисел {a₁, …, aₙ}, для которого выполняются равенства:
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + ( 1) + 1 + ( 1) + 5. All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13
A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k ( 1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. The product of this set of integers will yield n no matter how many 1s there are in the set. Furthermore, still under this assumption, any integer n would be amenable. Consider the inelegant solution for n of 1={1, 1, 1, 1, n}. In the sum, the positive ones are cancelled out by the negative ones, leaving n, while in the product, the two negative ones cancel out the effect of their signs. Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.
∑ᵢ aᵢ = n и ∏ᵢ aᵢ = n.
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + ( 1) + 1 + ( 1) + 5. All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13
A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k ( 1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. The product of this set of integers will yield n no matter how many 1s there are in the set. Furthermore, still under this assumption, any integer n would be amenable. Consider the inelegant solution for n of 1={1, 1, 1, 1, n}. In the sum, the positive ones are cancelled out by the negative ones, leaving n, while in the product, the two negative ones cancel out the effect of their signs. Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.
В мультимножестве допускаются отрицательные числа. Например, 5 является поддающимся, так как 5 = 1 + 1 + 1 + 1 + 1 + (-1) + (-1) + (-1) + (-1) + (-1). Все и только те числа, которые дают остаток 0 или 1 при делении на 4 (то есть, сравнимы с 0 или 1 по модулю 4), за исключением числа 4, являются поддающимися. Первые несколько поддающихся чисел: 1, 5, 8, 9, 12, 13.
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + ( 1) + 1 + ( 1) + 5. All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13
A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k ( 1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. The product of this set of integers will yield n no matter how many 1s there are in the set. Furthermore, still under this assumption, any integer n would be amenable. Consider the inelegant solution for n of 1={1, 1, 1, 1, n}. In the sum, the positive ones are cancelled out by the negative ones, leaving n, while in the product, the two negative ones cancel out the effect of their signs. Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.
Решение для целых чисел вида *n* = 4*k* + 1 можно получить, используя мультимножество, состоящее из 2*k* единиц, 2*k* минус единиц и самого числа *n*. (Это обобщает пример с числом 5, приведённый выше.) Хотя это и не очевидно из определения, множество поддающихся чисел замкнуто относительно умножения (произведение двух поддающихся чисел является поддающимся числом). Все составные числа были бы поддающимися, если бы мультимножеству разрешалось иметь любую длину, поскольку, даже если существуют другие решения, всегда можно получить решение, взяв разложение числа на простые множители (представленное повторением множителей, а не степеней) и добавив столько единиц, сколько необходимо, чтобы их сумма равнялась *n*. Произведение этого мультимножества целых чисел будет равно *n* независимо от количества единиц в нём. Более того, при этом предположении любое целое число *n* будет поддающимся. Рассмотрим неэлегантное решение для *n* = 1: {1, 1, 1, 1, 1}. В сумме положительные единицы компенсируются отрицательными, оставляя *n*, а в произведении два отрицательных числа компенсируют влияние своих знаков. Поддающиеся числа не следует путать с дружественными числами, которые представляют собой пары целых чисел, сумма делителей которых равна другому числу в паре.
hold. Negative numbers are allowed in the multiset. For example, 5 is amenable since 5 = 1 + ( 1) + 1 + ( 1) + 5. All and only those numbers congruent to 0 or 1 (mod 4), except 4, are amenable. The first few amenable numbers are: 1, 5, 8, 9, 12, 13
A solution for integers of the form n = 4k + 1 could be given by a set of 2k (+1)s and 2k ( 1)s and n itself. (This generalizes the example of 5 given above.) Although not obvious from the definition, the set of amenable numbers is closed under multiplication (the product of two amenable numbers is an amenable number). All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always obtain a solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n. The product of this set of integers will yield n no matter how many 1s there are in the set. Furthermore, still under this assumption, any integer n would be amenable. Consider the inelegant solution for n of 1={1, 1, 1, 1, n}. In the sum, the positive ones are cancelled out by the negative ones, leaving n, while in the product, the two negative ones cancel out the effect of their signs. Amenable numbers should not be confused with amicable numbers, which are pairs of integers whose divisors add up to each other.