Jaycob Coleman

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Let $\dfrac{\sigma(f_0)+h}{f_0}=\dfrac{\sigma(f_1)}{f_1}=\dfrac{\sigma(f_2)}{f_2}=\cdots=\dfrac{\sigma(f_s)}{f_s}$, where $\gcd(\sigma(f_0)+h,f_0)=1$, $h>0$, and $f_0<f_1<\cdots<f_s$. Let $r_i$ be the least positive integer such that every $1\leq m\leq(r_i-1)\sigma(f_i)$ has a representation as a sum of at most $r_i-1$ of each of the divisors of $f_i$. Conjecture: $r_0\geq r_1\geq\cdots\geq r_s$.

 9 Generalizations of $\sum_{m=3n+2}^{\infty}\phi^m=\phi^{3n}$ and $\sum_{m=13n+1}^{\infty}(\sqrt2-1)^m=\dfrac{(\sqrt2-1)^{13n}}{\sqrt2}$ 8 Prove that $\forall p \in \Bbb P;p \ne 5,$ $F_{p^n - \left(\frac{5}{p}\right)p^{n-1}} \equiv 0 \mod p^n$ 7 Does $A193201$ count the partitions of $n$ of arbitrary dimension? 7 Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number 6 Does there exist a positive $k$ s.t. for all $r\geq k$, “$\sigma_r(m)<\sigma_r(n)$ for every $m 257 Reputation  +5 Determine if$\sum_{q=1}^{\lceil n/2\rceil}R_q(n)$gives the number of divisors of$n$. +10 A similar, but hopefully easier problem than Gilbreath's conjecture +35 Does$A193201$count the partitions of$n$of arbitrary dimension? +5 Does there exist a sequence$(S_i)_{i=1}^{\infty},\ S_i=\pm1$such that$\forall i(2+S_1g_1+S_2g_2+\cdots+S_ig_i\in\Bbb P)\wedge\exists i:S_i=-1\$?

 34 Are there any open mathematical puzzles? 7 Is it sufficient to say that no odd divides an even number to prove it is a power of two? 5 Are there any open mathematical puzzles? 3 Best way to discover the 'golden ratio' 3 what is the pattern in the distribution of divisors.

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 9 elementary-number-theory × 12 3 golden-ratio × 2 9 number-theory × 4 3 normal-distribution 4 divisibility × 4 1 divisor-sum × 15 3 pattern-recognition × 2 1 proof-writing × 3 3 algebra-precalculus × 2 1 alternative-proof

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