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In mathematics, a ''k''-hyperperfect number is a natural number ''n'' for which the equality ''n'' = 1 + ''k''(''σ''(''n'') − ''n'' − 1) holds, where ''σ''(''n'') is the divisor function (i.e., the sum of all positive divisors of ''n''). A hyperperfect number is a ''k''-hyperperfect number for some integer ''k''. Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of ''k''-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of ''k'' being 1, 2, 1, 6, 3, 1, 12, ... . The first few ''k''-hyperperfect numbers that are not perfect are 21, 301, 325, 697, 1333, ... . ==List of hyperperfect numbers== The following table lists the first few ''k''-hyperperfect numbers for some values of ''k'', together with the sequence number in the On-Line Encyclopedia of Integer Sequences (OEIS) of the sequence of ''k''-hyperperfect numbers: It can be shown that if ''k'' > 1 is an odd integer and ''p'' = (3''k'' + 1) / 2 and ''q'' = 3''k'' + 4 are prime numbers, then ''p''²''q'' is ''k''-hyperperfect; Judson S. McCranie has conjectured in 2000 that all ''k''-hyperperfect numbers for odd ''k'' > 1 are of this form, but the hypothesis has not been proven so far. Furthermore, it can be proven that if ''p'' ≠ ''q'' are odd primes and ''k'' is an integer such that ''k''(''p'' + ''q'') = ''pq'' - 1, then ''pq'' is ''k''-hyperperfect. It is also possible to show that if ''k'' > 0 and ''p'' = ''k'' + 1 is prime, then for all ''i'' > 1 such that ''q'' = ''p''''i'' − ''p'' + 1 is prime, ''n'' = ''p''''i'' − 1''q'' is ''k''-hyperperfect. The following table lists known values of ''k'' and corresponding values of ''i'' for which ''n'' is ''k''-hyperperfect: 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hyperperfect number」の詳細全文を読む スポンサード リンク
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