Problem 57472. Compute the Lagarias Riemann Hypothesis sequence

Created by ChrisR

Solve Later

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<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 132px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 66px; transform-origin: 407px 66px; vertical-align: baseline; "><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 135.225px 8px; transform-origin: 135.225px 8px; unicode-bidi: normal; ">Write a function that takes an input number n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 203.45px 8px; transform-origin: 203.45px 8px; unicode-bidi: normal; "> and produces a sequence (i.e., all values up to and including the n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7833px 8px; transform-origin: 28.7833px 8px; unicode-bidi: normal; ">th value) computed with </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><img 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" alt="a(n) = floor(H(n) + exp(H(n))ln(H(n)))-sigma(n)" style="width: 280px; height: 18.5px;" width="280" height="18.5"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21.0083px 8px; transform-origin: 21.0083px 8px; unicode-bidi: normal; ">where <img src="data:image/png;base64,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" alt="H(n)" style="width: 35.5px; height: 18.5px;" width="35.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; "> is the n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.775px 8px; transform-origin: 7.775px 8px; unicode-bidi: normal; ">th <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46000">harmonic number</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" alt="sigma(n)" style="width: 30.5px; height: 18.5px;" width="30.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20.6083px 8px; transform-origin: 20.6083px 8px; unicode-bidi: normal; "> is the <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46898">sum of divisors</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 9.71667px 8px; transform-origin: 9.71667px 8px; unicode-bidi: normal; "> of n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; ">. </div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.7917px 8px; transform-origin: 28.7917px 8px; unicode-bidi: normal; ">Lagarias <a target='_blank' href = "https://arxiv.org/abs/math/0008177">proved</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.55px 8px; transform-origin: 15.55px 8px; unicode-bidi: normal; "> that <img src="data:image/png;base64,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" alt="a(n) >= 0" style="width: 56px; height: 18.5px;" width="56" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 21px 8px; transform-origin: 21px 8px; unicode-bidi: normal; "> for all n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 33.8333px 8px; transform-origin: 33.8333px 8px; unicode-bidi: normal; "> only if the <a target='_blank' href = "https://en.wikipedia.org/wiki/Riemann_hypothesis">Riemann Hypothesis</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 24.8833px 8px; transform-origin: 24.8833px 8px; unicode-bidi: normal; "> is true. </div></div></div>

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Last Solution submitted on Dec 31, 2022

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2 Comments

William on 24 Dec 2022

Hi Chris. I think a couple of changes need to be made in the problem description. First, there is a sign error in the definition of a(n). Second, it should say that the function, with argument 'n', is supposed to return an array of values a(1:n) and not just a single value.

ChrisR on 25 Dec 2022

Thanks William. I made those changes.

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Problem 57472. Compute the Lagarias Riemann Hypothesis sequence

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Problem 57472. Compute the Lagarias Riemann Hypothesis sequence

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