Lien-Hung Su

On Divisibility-Constrained Partitions of an Arbitrary Integer

We consider integer partitions for which the sum of the parts divides their product. Five OEIS counting functions are examined: the fixed-four-part count A343320 and the unrestricted-length counts A057568, A379733, A379734, and A379735, with distinctness and exclusion of the part...

Universal and Simultaneous Constructions of Perfect Powers with Equal Decimal Digit Frequencies

We construct perfect powers whose decimal representations contain each of the ten digits equally often. A general even-exponent construction is obtained from full-reptend prime-power denominators and a digit-preserving cancellation of signed block carries. The construction is unc...

Palfree Squares in a Two-Parameter Family of Near-Repdigits

For integers \(1 \le d \le 9\), \(0 \le e \le 9\), and \(n \ge 1\), let\[N_{d,e}(n)=d\frac{10^n-1}{9}+(e-d).\]For \(n \ge 2\), this is the decimal integer consisting of \(n-1\) copies of \(d\) followed by \(e\). We classify exactly when \(N_{d,e}(n)^2\) is palfree, meaning that i...

Adjacent-Digit Distinctness in Squares of Generalized Repunits

For an integer base \(b\ge 3\), let \[ R_n^{(b)}=1+b+\cdots+b^{n-1}=\frac{b^n-1}{b-1} \] be the length-\(n\) repunit in base \(b\). We prove that the standard base-\(b\) expansion of \(\bigl(R_n^{(b)}\bigr)^2\) has no two equal adjacent digits if and only if \(n=1\) or \(n\not\eq...

Explicit Infinite Families of Squares, Cubes, and Fourth Powers with No Equal Adjacent Decimal Digits

We study three fixed-exponent classes separately: squares, corresponding to OEIS A050741/A050749; cubes, corresponding to A050742/A050750; and fourth powers, corresponding to A050743/A050751. For squares, we recast the known construction of Rodrigo (2017), based on the roots \(R_...

Explicit Infinite Families of Squares, Cubes, and Fourth Powers with No Equal Adjacent Decimal Digits

We study three fixed-exponent classes separately: squares, corresponding to OEIS A050741/A050749; cubes, corresponding to A050742/A050750; and fourth powers, corresponding to A050743/A050751. For squares, we recast the known construction of Rodrigo (2017), based on the roots \(R_...

Infinitude of a Greedy Sequence of Perfect Powers with Distinct Neighboring Digits

We study the increasing greedy construction represented by OEIS A175031. Starting from perfect powers whose neighboring decimal digits are distinct, after a term is chosen the next term is the least larger admissible perfect power whose first digit differs from the last digit of...

An Explicit Infinite Family of Fifth Powers with No Equal Adjacent Decimal Digits

We study OEIS A050744/A050752, where A050744 records integers \(x\) for which the decimal expansion of \(x^5\) has no pair of equal adjacent digits and A050752 records the corresponding fifth powers. For every integer \(t\ge1\), we consider \[ x_t=\frac{2(10^{54t}-5\cdot10^{27t}+...

Adjacent-Digit Distinctness in Squares of Generalized Repunits

For an integer base \(b \ge 3\), let\[R_n^{(b)}=1+b+\cdots+b^{n-1}=\frac{b^n-1}{b-1}\]be the length-\(n\) repunit in base \(b\). We prove that the standard base-\(b\) expansion of\[\left(R_n^{(b)}\right)^2\]has no two equal adjacent digits if and only if\[n=1\qquad\text{or}\qquad...

Explicit Square Constructions for Odd-Length Geometric Progressions

OEIS A092960 records the least square representable as one plus the sum of $n$ successive terms of a geometric progression, with value $0$ when no such square exists. For every odd $n\ge3$, we give an explicit $n$-term geometric progression with first term $2^n-3$ and common rati...

Nonvanishing and Growth of Least Prime Denominators for Decimal Powers

This preprint studies OEIS A090519, where \(a(n)\) is the least prime \(p\) such that \(\lfloor 10^n/p\rfloor\) is prime, with value 0 if no such prime exists. We prove that \(a(n)>0\) for every \(n\ge 1\), resolving the nonvanishing conjecture recorded for the sequence. An eleme...

Nonvanishing and Growth of Least Prime Denominators for Decimal Powers

This preprint studies OEIS A090519, where \(a(n)\) is the least prime \(p\) such that \(\lfloor 10^n/p\rfloor\) is prime, with value 0 if no such prime exists. We prove that \(a(n)>0\) for every \(n\ge 1\), resolving the nonvanishing conjecture recorded for the sequence. An eleme...

Prime Witnesses in a Floor-Quotient Sequence

For an integer \(X\ge 1\), let \(\mathcal W(X)\) be the set of primes \(p\) for which \(\lfloor X/p\rfloor\) is prime, and let \(W(X)=|\mathcal W(X)|\). For a fixed prime \(q\), we prove the exact identity\[W_q(X)=\pi(X/q)-\pi(X/(q+1)),\]and hence\[W_q(X)\sim \frac{X}{q(q+1)\log...

Arbitrarily Long Increasing and Decreasing Runs in ⌊n^(p_n/n)⌋

Let \(p_n\) denote the \(n\)th prime and define\[A_n=\left\lfloor n^{p_n/n}\right\rfloor.\]OEIS A318199 records the conjecture that no consecutive strictly increasing run contains more than 17 terms. Introducing \(B_n=n^{p_n/n}\) and \(g_n=p_{n+1}-p_n\), we derive the exact ident...

Arbitrarily Long Increasing and Decreasing Runs in ⌊n^(p_n/n)⌋

Let \(p_n\) denote the \(n\)th prime and define\[A_n=\left\lfloor n^{p_n/n}\right\rfloor.\]OEIS A318199 records the conjecture that no consecutive strictly increasing run contains more than 17 terms. Introducing \(B_n=n^{p_n/n}\) and \(g_n=p_{n+1}-p_n\), we derive the exact ident...

The Read-Once Factorial-Divisibility Conjecture: Algebraic Proofs Through Six Inputs

We formulate the division-free read-once factorial-divisibility conjecture for pairwise distinct integers. Using congruence arguments, signed residue classes, parity decompositions, and disjoint block constructions, we prove the conjecture algebraically for all input sizes 1≤n≤6....

The Read-Once Factorial-Divisibility Conjecture: Algebraic Proofs Through Six Inputs

We formulate the division-free read-once factorial-divisibility conjecture for pairwise distinct integers. Using congruence arguments, signed residue classes, parity decompositions, and disjoint block constructions, we prove the conjecture algebraically for all input sizes 1≤n≤6....

The Read-Once Factorial-Divisibility Conjecture: Algebraic Proofs Through Six Inputs

We formulate the division-free read-once factorial-divisibility conjecture for pairwise distinct integers. Using congruence arguments, signed residue classes, parity decompositions, and disjoint block constructions, we prove the conjecture algebraically for all input sizes 1≤n≤6....