Numbers in the base $e^π$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914047034654720 |
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| author | Plouffe, Simon |
| author_facet | Plouffe, Simon |
| contents | A large-scale experiment was conducted to find formulas relating to the base $e^π$. The numbers in this base are $$x = \sum_{n=0}^\infty {a(n)\over e^{πn}}$$ where $a(n)$ is taken from the OEIS catalog. These experiments were inspired by several facts. Indeed, it is known that the formula generating the partitions of integers is generated by an infinite product $$\prod_{k\ge1}^\infty {1\over 1-x^k} = \sum_{n=0}^\infty p(n)x^n$$ that when evaluated at $x=e^{-π}$ is equal to $${2^{3/8} Γ(3/4) \over π^{1/4} e^{π/24}}\ .\qquad\qquad (1)$$ By analyzing the 387500 sequences of the OEIS catalog, the model that was used is based on the fact that the infinite sum evaluated at $e^π$, is an expression that can be detected using a program like lindep from Pari-Gp. The process made it possible to find 793 expresssions similar to (1). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15609 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Numbers in the base $e^π$ Plouffe, Simon Number Theory Primary 11Y40, Secondary 11Y55 A large-scale experiment was conducted to find formulas relating to the base $e^π$. The numbers in this base are $$x = \sum_{n=0}^\infty {a(n)\over e^{πn}}$$ where $a(n)$ is taken from the OEIS catalog. These experiments were inspired by several facts. Indeed, it is known that the formula generating the partitions of integers is generated by an infinite product $$\prod_{k\ge1}^\infty {1\over 1-x^k} = \sum_{n=0}^\infty p(n)x^n$$ that when evaluated at $x=e^{-π}$ is equal to $${2^{3/8} Γ(3/4) \over π^{1/4} e^{π/24}}\ .\qquad\qquad (1)$$ By analyzing the 387500 sequences of the OEIS catalog, the model that was used is based on the fact that the infinite sum evaluated at $e^π$, is an expression that can be detected using a program like lindep from Pari-Gp. The process made it possible to find 793 expresssions similar to (1). |
| title | Numbers in the base $e^π$ |
| topic | Number Theory Primary 11Y40, Secondary 11Y55 |
| url | https://arxiv.org/abs/2509.15609 |