Problem 46582. Find jumping medalists

Created by ChrisR

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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: 288px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 144px; transform-origin: 407.5px 144px; vertical-align: baseline; "><div style="block-size: 63px; 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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; 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: 377.308px 7.875px; transform-origin: 377.308px 7.875px; unicode-bidi: normal; "><span style="">Key questions in number theory involve the distribution of prime numbers. For example, the Twin Prime Conjecture states that infinitely many twin primes, or two primes separated by 2, exist. This conjecture has not been proved, and progress is addressed in an interesting </span></span><a target='_blank' href = "https://www.youtube.com/watch?v=vkMXdShDdtY"><span style=""><span style="">video from Numberphile</span></span></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: 3.88333px 7.875px; transform-origin: 3.88333px 7.875px; unicode-bidi: normal; "><span style="">. </span></span></div><div style="block-size: 63px; 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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; 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: 374.983px 7.875px; transform-origin: 374.983px 7.875px; unicode-bidi: normal; "><span style="">This problem deals with the most common gap between primes up to a given number. John Conway dubbed this gap the </span></span><a target='_blank' href = "https://mathworld.wolfram.com/JumpingChampion.html"><span style=""><span style="perspective-origin: 56.8083px 7.875px; transform-origin: 56.8083px 7.875px; ">jumping champion</span></span></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: 324.392px 7.875px; transform-origin: 324.392px 7.875px; unicode-bidi: normal; "><span style="">. For numbers up to 20, the jumping champion is 2 because it occurs four times (between 3 and 5, 5 and 7, 11 and 13, and 17 and 19.) </span></span></div><div style="block-size: 63px; 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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 31.5px; 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: 293.708px 7.875px; transform-origin: 293.708px 7.875px; unicode-bidi: normal; "><span style="">To me, the jumping champion is somewhat disappointing because 6 dominates until about 1.74</span></span><span style="vertical-align:-5px"><img src="data:image/png;base64,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" style="width: 37.5px; height: 19px;" width="37.5" height="19"></span><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: 68.45px 7.875px; transform-origin: 68.45px 7.875px; unicode-bidi: normal; "><span style="">. Therefore, I will coin another term: the </span></span><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: 56.025px 7.875px; transform-origin: 56.025px 7.875px; unicode-bidi: normal; "><span style="font-style: italic; ">jumping medalists</span></span><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: 266.817px 7.875px; transform-origin: 266.817px 7.875px; unicode-bidi: normal; "><span style="">, or the three most common gaps between primes up to a given number. For numbers up to 20, the gold, silver, and bronze jumping medals (i.e., first, second, and third place) go to 2, 4, and 1, respectively.</span></span></div><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: 384.5px 21px; text-align: left; transform-origin: 384.5px 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: 336.983px 7.875px; transform-origin: 336.983px 7.875px; unicode-bidi: normal; "><span style="">Write a function that determines the jumping medalists as well as the maximum gap. Award the medals as in </span></span><a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/46576-award-medals-to-winners"><span style=""><span style="">Cody Problem 46576</span></span></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: 206.925px 7.875px; transform-origin: 206.925px 7.875px; unicode-bidi: normal; "><span style=""> and return an empty vector for any medal that cannot be awarded.</span></span></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: 384.5px 10.5px; text-align: left; transform-origin: 384.5px 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: 0px 7.875px; transform-origin: 0px 7.875px; unicode-bidi: normal; "><span style=""></span></span></div></div></div>

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Last Solution submitted on Aug 06, 2021

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Jean-Marie Sainthillier on 3 Apr 2021

In test6 with n=100, J3=1 no ?

ChrisR on 5 Apr 2021

Jean-Marie, in Test 6, three medals (1 gold, 2 silver) have already been awarded. By the rules from Cody Problem 46576, no bronze is awarded.

Jean-Marie Sainthillier on 5 Apr 2021

OK.
Thanks.

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Problem 46582. Find jumping medalists

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