A number n is practical if every integer in [1, n] can be expressed as a subset sum of the positive divisors of n. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime k-tuples conjecture for practical numbers, proving the “correct” upper bound, and for pairs, improving on a lower bound of Melfi.

中文翻译：

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关于素数和实数

如果[1，n ]中的每个整数 都可以表示为n的正因数的子集和，则数n是实用的。我们考虑实际数字的分布，这些分布也移动了质数，从而改进了Guo和Weingartner的一个定理。此外，从本质上证明马根斯特恩猜想，我们证明了所有大的奇数都是素数和实际数的和。对于实数，我们还考虑了素数k元组猜想的类似物，证明了“正确”的上限，对数对，则提高了Melfi的下限。