斐波那契序列与卢卡斯序列中的史密斯数：一项计算探索
When Fibonacci and Lucas Meet Smith: A Computational Exploration
- Universidade Tecnológica Federal do Paraná (UTFPR)（巴西联邦技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过计算搜索，在斐波那契序列和卢卡斯序列中发现新的史密斯数，其中卢卡斯序列在索引小于1000时找到10个，斐波那契序列找到7个，且两序列在索引827处均存在史密斯数。
AI中文摘要：
斐波那契序列和卢卡斯序列是趣味数论中的经典序列。本文提出一个简单问题：它们的项 $F_n$ 和 $L_n$ 中哪些是史密斯数？史密斯数是指一个合数，其十进制各位数字之和等于其质因数（按重数计）的十进制各位数字之和。该问题表述简单，但对于较大的 $n$，很快会转化为一个因式分解问题。利用已有的完全因式分解数据，我们对两个序列开展了计算搜索，在每个序列中都得到了新的史密斯项。对于卢卡斯序列，第一作者的搜索在索引小于1000时发现了10个史密斯项：3、95、105、114、183、437、609、682、827、902。这些结果已作为OEIS A395686发表在《在线整数序列百科全书》中，随后Sean A. Irvine又为该序列添加了4个索引：1090、1153、1215、1378。对于斐波那契序列，从OEIS A382922中已记录的案例出发，我们的搜索得到了另外7个史密斯数：$F_{1440}$、$F_{1554}$、$F_{1596}$、$F_{1863}$、$F_{2256}$、$F_{2277}$、$F_{2559}$。对于偶索引的情况，恒等式 $F_{2n}=F_nL_n$ 允许将已有的斐波那契和卢卡斯因式分解数据结合起来，以获取史密斯检验所需的完全因式分解。令人惊讶的是，两次搜索在索引827处交汇：$F_{827}$ 和 $L_{827}$ 均为史密斯数。
英文摘要：
The Fibonacci and Lucas sequences are old friends in recreational number theory. Here we ask a simple question: which of their terms $F_n$ and $L_n$ are Smith numbers? A Smith number is a composite integer whose decimal digit sum equals the sum of the decimal digits of its prime factors, counted with multiplicity. The question is easy to state, but for large $n$ it quickly becomes a factorization problem. Using available complete factorization data, we carried out computational searches in both sequences and obtained new Smith terms in each of them. For the Lucas sequence, the first author's search found ten Smith terms with indices below 1000: 3, 95, 105, 114, 183, 437, 609, 682, 827, 902. These results were published in the On-Line Encyclopedia of Integer Sequences as OEIS A395686. Four further indices, 1090, 1153, 1215, 1378, were subsequently added to the sequence by Sean A. Irvine. For the Fibonacci sequence, starting from the cases already recorded in OEIS A382922, our search produced seven further Smith numbers: $F_{1440}$, $F_{1554}$, $F_{1596}$, $F_{1863}$, $F_{2256}$, $F_{2277}$, $F_{2559}$. For the even-index cases, the identity $F_{2n}=F_nL_n$ allows available Fibonacci and Lucas factorization data to be combined to recover the complete factorizations needed for the Smith test. Surprisingly, the two searches also meet at index 827: both $F_{827}$ and $L_{827}$ are Smith numbers.
