Three Cousins of Recaman's Sequence
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arXiv
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| Natura: | Preprint |
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2020
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| _version_ | 1866910802740510720 |
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| author | Alekseyev, Max A. Myers, Joseph Samuel Schroeppel, Richard Shannon, S. R. Sloane, N. J. A. Zimmermann, Paul |
| author_facet | Alekseyev, Max A. Myers, Joseph Samuel Schroeppel, Richard Shannon, S. R. Sloane, N. J. A. Zimmermann, Paul |
| contents | Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and add n+1, n+2, n+3, ..., stopping after adding n+k if the sum n + (n+1) + ... + (n+k) is divisible by n+k+1. Then A(n)=k. We determine A(n) and show that A(n) <= n^2 - 2n - 1. (ii) {B(n), n >= 1} is a multiplicative analog of {A(n)}. Start with n, and successively multiply by n+1, n+2, ..., stopping after multiplying by n+k if the product n(n+1)...(n+k) is divisible by n+k+1. Then B(n)=k. We conjecture that log^2 B(n) = (1/2 + o(1)) log n loglog n. (iii) The third sequence, {C(n), n >= 1}, is the most interesting, because the most mysterious. Concatenate the decimal digits of n, n+1, n+2, ... until the concatenation n||n+1||...||n+k is divisible by n+k+1. Then C(n)=k. If no such k exists we set C(n)=-1. We have found k for all n <= 1000 except for two cases. Some of the numbers involved are quite large. For example, C(92) = 218128159460, and the concatenation 92||93||...||(92+C(92)) is a number with about 2*10^12 digits. We have only a probabilistic argument that such a k exists for all n. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2004_14000 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Three Cousins of Recaman's Sequence Alekseyev, Max A. Myers, Joseph Samuel Schroeppel, Richard Shannon, S. R. Sloane, N. J. A. Zimmermann, Paul Number Theory 11B83 (11D72, 11D85) Although 10^230 terms of Recaman's sequence have been computed, it remains a mystery. Here three distant cousins of that sequence are described, one of which is also mysterious. (i) {A(n), n >= 3} is defined as follows. Start with n, and add n+1, n+2, n+3, ..., stopping after adding n+k if the sum n + (n+1) + ... + (n+k) is divisible by n+k+1. Then A(n)=k. We determine A(n) and show that A(n) <= n^2 - 2n - 1. (ii) {B(n), n >= 1} is a multiplicative analog of {A(n)}. Start with n, and successively multiply by n+1, n+2, ..., stopping after multiplying by n+k if the product n(n+1)...(n+k) is divisible by n+k+1. Then B(n)=k. We conjecture that log^2 B(n) = (1/2 + o(1)) log n loglog n. (iii) The third sequence, {C(n), n >= 1}, is the most interesting, because the most mysterious. Concatenate the decimal digits of n, n+1, n+2, ... until the concatenation n||n+1||...||n+k is divisible by n+k+1. Then C(n)=k. If no such k exists we set C(n)=-1. We have found k for all n <= 1000 except for two cases. Some of the numbers involved are quite large. For example, C(92) = 218128159460, and the concatenation 92||93||...||(92+C(92)) is a number with about 2*10^12 digits. We have only a probabilistic argument that such a k exists for all n. |
| title | Three Cousins of Recaman's Sequence |
| topic | Number Theory 11B83 (11D72, 11D85) |
| url | https://arxiv.org/abs/2004.14000 |