Problem 52669. Count the primes in Collatz sequences

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: 165px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 82.5px; transform-origin: 407px 82.5px; vertical-align: baseline; "><div style="block-size: 84px; 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 42px; text-align: left; transform-origin: 384px 42px; 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: 78px 8px; transform-origin: 78px 8px; unicode-bidi: normal; ">Several Cody problems (<a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/21">21</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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/69">69</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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/42937">42937</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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/44448">44448</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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/44784">44784</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: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">, <a target='_blank' href = "https://www.mathworks.com/matlabcentral/cody/problems/52422">52422</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: 165.5px 8px; transform-origin: 165.5px 8px; unicode-bidi: normal; ">) involve Collatz sequences. These start with a seed 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: 9.5px 8px; transform-origin: 9.5px 8px; unicode-bidi: normal; ">. If 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: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; "> is odd, the next element is <img src="data:image/png;base64,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" alt="3n+1" style="width: 42.5px; height: 18px;" width="42.5" height="18"><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: 23.5px 8px; transform-origin: 23.5px 8px; unicode-bidi: normal; ">, and if 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: 89.5px 8px; transform-origin: 89.5px 8px; unicode-bidi: normal; "> is even, the next element is <img src="data:image/png;base64,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" alt="n/2" style="width: 25.5px; height: 18.5px;" width="25.5" 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: 125px 8px; transform-origin: 125px 8px; unicode-bidi: normal; ">. For example, if the seed is 3, then the sequence is 3, 10, 5, 16, 8, 4, 2, 1. The Collatz conjecture is that all of these sequences terminate at the value 1; that is, all seeds lead to terminating sequences. </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: 366px 8px; transform-origin: 366px 8px; unicode-bidi: normal; ">This problem deals with the number of primes in the sequence. With a seed of 3, the number of primes is 3 (2, 3, 5). </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: 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: 375.5px 8px; transform-origin: 375.5px 8px; unicode-bidi: normal; ">Write a function to determine the numbers of primes in the sequences with seeds of 1 to the input number. See the test suite for banned terms and commands. </div></div></div>

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Last Solution submitted on May 04, 2023

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Tim on 8 Sep 2021

I notice that p=isprime(x) can be a lot slower than p=ismember(x,primes(max(x))); for x=1:1e7, the former takes about 100 seconds and the latter takes about 0.1 seconds.

ChrisR on 11 Sep 2021

Nice observation. I'll keep that one in mind.

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Problem 52669. Count the primes in Collatz sequences

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