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Question Number 168885 by Rasheed.Sindhi last updated on 20/Apr/22

$$$$$$\mathrm{Find}\mathrm{all}\mathrm{the}\mathrm{values}\mathrm{of}\mathrm{n}\mathrm{such}\mathrm{that}:$$$$\overline{)\begin{array}{c}\mathrm{digit}-\mathrm{sum}\left({\mathrm{n}}^3\right)=\mathrm{n}\end{array}}$$$$\mathrm{and}$$$$\mathrm{show}\mathrm{that}\mathrm{n}\mathrm{has}\mathrm{at}\mathrm{most}2\mathrm{digits}.$$$${}^{◼}\mathrm{digit}-\mathrm{sum}\left(\overline{\mathrm{abc}..}\right)=\mathrm{a}+\mathrm{b}+\mathrm{c}+...$$

Commented by botir last updated on 20/Apr/22

$$verynice$$

Commented by Rasheed.Sindhi last updated on 20/Apr/22

$$\mathcal{T}han\mathcal{X}sir!$$

Commented by MJS_new last updated on 20/Apr/22

$$n=0\mathrm{and}\mathrm{n}=1\mathrm{are}\mathrm{solutions}$$$$\mathrm{so}\mathrm{we}\mathrm{search}\mathrm{for}n\in \mathbb{N}\wedge n>1$$

Commented by MJS_new last updated on 20/Apr/22

$$0=0^3$$$$1=1^3$$$$512=8^3$$$$4913={17}^3$$$$5832={18}^3$$$$17576={26}^3$$$$19683={27}^3$$$$\mathrm{no}\mathrm{more}\mathrm{solutions}$$$$\mathrm{the}\mathrm{sum}\mathrm{of}\mathrm{a}6-\mathrm{digit}\mathrm{number}\mathrm{is}⩽54$$$$\Rightarrow \mathrm{no}\mathrm{possible}\mathrm{solution}>54$$

Commented by MJS_new last updated on 21/Apr/22

$$\mathrm{for}\mathrm{\Sigma }\mathrm{digits}\left(n^k\right)=n\mathrm{with}k\in {\mathbb{N}}^{★}\mathrm{we}\mathrm{have}\mathrm{to}$$$$\mathrm{try}\mathrm{all}n\in {\mathbb{N}}^{★}\left(0\mathrm{and}1\mathrm{are}\mathrm{always}\mathrm{solutions}\right)$$$$\mathrm{with}$$$$1<n<z$$$$\mathrm{with}z\mathrm{being}\mathrm{the}\mathrm{greatest}\mathrm{zero}\mathrm{of}$$$$f\left(z\right)=9\times (1+\lfloor k{\log }_{10}n\rfloor )-n$$

Commented by Rasheed.Sindhi last updated on 20/Apr/22

$$\mathrm{Great}!$$$$\mathbb{T}\mathbf{han}\mathbb{k}\mathbf{s}\mathbf{sir}!$$

Commented by Rasheed.Sindhi last updated on 21/Apr/22

$$\mathrm{\Sigma }\mathrm{digits}\left(n^k\right)={\left(\mathrm{\Sigma }\mathrm{digits}\left(n\right)\right)}^k?$$$$Sirmyquestionwasabout$$$$\mathrm{\Sigma }\mathrm{digits}\left(n^k\right).$$

Commented by MJS_new last updated on 21/Apr/22

$$\mathrm{sorry}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{a}\mathrm{typo}.\mathrm{I}\mathrm{corrected}\mathrm{it}.$$

Commented by Rasheed.Sindhi last updated on 21/Apr/22

$$\mathcal{T}han\mathcal{X}\mathcal{S}ir!$$$$Aretheseformulasresultof$$$$\mathit{g}\mathit{o}\mathit{o}\mathit{g}\mathit{l}\mathit{i}\mathit{n}\mathit{g}?$$

Commented by MJS_new last updated on 21/Apr/22

no. brainwork. I used a simple calculator and started with n=1, 2, 3, ... then I found that there must be a border because the max sum of k digits is 9k...

Commented by Rasheed.Sindhi last updated on 21/Apr/22

No doubt your calculator is simple but your brain is as complex as mathematics is. No doubt your formulae are creative work sir!

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