Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0, \[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \] and for k ≥ 0, \[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \] where spt(n) denotes Andrews's smallest parts function.

中文翻译：

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纽曼恒等式、卢卡斯序列和某些配分函数的同余

让r为 2 ≤ 的整数r≤ 24 并让pr(n) 定义为$\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. 在本文中，我们提供了发现无限同余族和奇异同余的统一方法pr(n) 通过使用一些身份pr(n) 由于纽曼。作为应用，我们为某些配分函数建立了许多无限的同余族和奇异同余族，例如 Andrews 的最小部分函数、Ramanujan 的系数φ功能和p- 正则分区函数。例如，我们证明对于n≥0，\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2 } +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]并且对于ķ≥0，\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]其中 spt(n) 表示安德鲁斯的最小部分函数。