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Question Number 222974 by gabthemathguy25 last updated on 12/Jul/25

Answered by MrGaster last updated on 12/Jul/25

P_(100) =min{x∈N∣Σ_(k=1) ^x Π_(i=2) ^(⌊(√k)⌋) (1−δ(k mod i))≥101} Σ_(k=1) ^(540) Π_(i=2) ^(⌊(√k)⌋) (1−δ(k mod i))=100<101 Σ_(k=1) ^(541) Π_(i=2) ^(⌊(√k)⌋) (1−δ(k mod i))=101≥101 P_(100) =541

$${P}_{\mathrm{100}} =\mathrm{min}\left\{{x}\in\mathbb{N}\mid\underset{{k}=\mathrm{1}} {\overset{{x}} {\sum}}\underset{{i}=\mathrm{2}} {\overset{\lfloor\sqrt{{k}}\rfloor} {\prod}}\left(\mathrm{1}−\delta\left({k}\:\mathrm{mod}\:{i}\right)\right)\geq\mathrm{101}\right\} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{540}} {\sum}}\underset{{i}=\mathrm{2}} {\overset{\lfloor\sqrt{{k}}\rfloor} {\prod}}\left(\mathrm{1}−\delta\left({k}\:\mathrm{mod}\:{i}\right)\right)=\mathrm{100}<\mathrm{101} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{\mathrm{541}} {\sum}}\underset{{i}=\mathrm{2}} {\overset{\lfloor\sqrt{{k}}\rfloor} {\prod}}\left(\mathrm{1}−\delta\left({k}\:\mathrm{mod}\:{i}\right)\right)=\mathrm{101}\geq\mathrm{101} \\ $$$${P}_{\mathrm{100}} =\mathrm{541} \\ $$

Commented by gabthemathguy25 last updated on 12/Jul/25

perfection!

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