Aperiodic Monotonic Characteristic Series

SPECIES

Monotonic characteristic series of triangular numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi\left(\dfrac{n(n+1)}{2}\right) = -\dfrac{1}{6}$ (unstable)

$1+0+1+0+0+1+0+0+0+1+\cdots = -\dfrac{1}{6}$ (unstable)

Monotonic characteristic series of square numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi(n^2) = \dfrac{1}{6}$ (unstable)

$1+0+0+1+0+0+0+0+1+\cdots = \dfrac{1}{6}$ (unstable)

Monotonic characteristic series of pentagonal numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi\left(\dfrac{n(3n-1)}{2}\right) = \dfrac{1}{2}$ (unstable)

$1+0+0+0+1+0+0+0+0+0+0+1+\cdots = \dfrac{1}{2}$ (unstable)

Monotonic characteristic series of hexagonal numbers (starting from 1)

$\displaystyle \sum_{n\geq1} \chi\left(n(2n-1)\right) = \dfrac{5}{6}$ (unstable)

$1+0+0+0+0+1+0+0+0+0+0+0+0+0+1+\cdots = \dfrac{5}{6}$ (unstable)

Monotonic characteristic series of powers of 2 (starting from 1)

$\displaystyle \sum_{n\geq1} \chi(2^n) = 1$ (stable)

$1+1+0+1+0+0+0+1+0+0+0+0+0+0+0+1+\cdots = 1$ (stable)

Monotonic characteristic series of prime numbers

$\displaystyle \sum_{p\in\mathbb{P}} \chi(p) = B_1$ (stable)

$0+1+1+0+1+0+1+0+0+0+1+0+1+\cdots = B_1^{-} = -\dfrac{1}{2}$ (stable)

$1+1+1+0+1+0+1+0+0+0+1+0+1+\cdots = B_1^{+} = \dfrac{1}{2}$ (stable)

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