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September 2023 - Challenge

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This riddle was proposed by Marco Bellocchi - thanks Marco!

Define a sequence a_1, a_2, a_3 recursively by setting a_n = a_{n-1}+\text{gcd}(n, a_{n-1}) where \text{gcd}(x, y) denotes the greatest common divisor (GCD) of the pair of numbers x, y.
For example, when a_1 = 11 we have a_2=12 since \text{gcd}(2,11)=1, and we have a_3=15 because \text{gcd(3,12)=3}.
Continuing, we arrive at the sequence:

11, 12, 15, 16, 17, 18, 19, 20, 21, 22, 33, 36,...
We now look at the sequence of differences beginning from n=2, d_n=a_n-a_{n-1}=\text{gcd}(n, a_{n-1}):
1, 3, 1, 1, 1, 1, 1, 1, 1, 11, 3, ...
(i.e., d_2=1 is the first element of the sequence, d_3=3 is the second, etc.)

A curious property enjoyed by this sequence is that it contains only the number 1 and the prime numbers 3, 11. If we continue this sequence long enough and erase all the occurrences of 1, we arrive at this sequence
3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3, 7, 1889, ...
which contains only prime numbers. It can be proven that indeed, when a_1=11, all elements of the sequence obtained in this manner will be prime.

Returning to the original difference sequence, it is easy to see that 3=d_{3}=d_{12}=d_{24}=d_{48}=d_{51} and these are the first 5 occurrences of 3 in the sequence. Hence, 3 appears for the fifth time for n=51.

Your goal: For the sequence defined by a_1=531, find the value of n for which 5 appears for the tenth time.

In addition, find values k, n such that for the sequence defined by the initial value a_1=k, we have d_n>1 but not a prime.

A Bonus "*" will be given for finding, for the sequence defined by a_1=531, the value of n for which 5 appears for the 200th time.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com


Challenge: 28/08/2023 @ 15:30 PM EST
Solution: 05/10/2023 @ 12:00 PM EST
List Updated: 06/11/2023 @ 12:10 PM EST

People who answered correctly:

Lazar Ilic(30/8/2023 2:35 PM IDT)
Daniel Chong Jyh Tar(30/8/2023 4:41 PM IDT)
Alex Fleischer(30/8/2023 6:04 PM IDT)
Grant Boudreaux(30/8/2023 6:20 PM IDT)
Paul Revenant(30/8/2023 8:30 PM IDT)
*Reda Kebbaj(30/8/2023 11:14 PM IDT)
*Alper Halbutogullari(30/8/2023 11:15 PM IDT)
*Franciraldo Cavalcante(31/8/2023 12:38 AM IDT)
Kipp Johnson(31/8/2023 2:41 AM IDT)
*Bertram Felgenhauer(31/8/2023 2:53 AM IDT)
Prashant Wankhede(31/8/2023 6:17 AM IDT)
*Giorgos Kalogeropoulos(31/8/2023 11:06 AM IDT)
Xiao Liu(31/8/2023 11:07 AM IDT)
Christofer Ohlsson(31/8/2023 12:05 PM IDT)
John Tromp(31/8/2023 12:30 PM IDT)
*Sanandan Swaminathan(31/8/2023 1:06 PM IDT)
Ralf Jonas(31/8/2023 3:36 PM IDT)
*Yan-Wu He(31/8/2023 6:38 PM IDT)
*Ariel Landau(31/8/2023 11:11 PM IDT)
*Andrew Gauld(1/9/2023 8:58 AM IDT)
Dieter Beckerle(1/9/2023 10:27 AM IDT)
*Vladimir Volevich(1/9/2023 11:40 AM IDT)
Thomas Egense(1/9/2023 1:37 PM IDT)
Jon Namnath(1/9/2023 2:50 PM IDT)
Fabio Michele Negroni(1/9/2023 3:09 PM IDT)
*Lorenzo Gianferrari Pini(1/9/2023 3:16 PM IDT)
Florian Sikora(1/9/2023 3:53 PM IDT)
Guillaume Escamocher(1/9/2023 3:59 PM IDT)
*Lorenz Reichel(1/9/2023 6:09 PM IDT)
Erik Wünstel(1/9/2023 7:13 PM IDT)
Blaine Hill(1/9/2023 7:48 PM IDT)
*Latchezar Christov(1/9/2023 9:50 PM IDT)
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Evan Semet(2/9/2023 7:49 AM IDT)
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Reiner Martin(3/9/2023 2:57 PM IDT)
*Hansraj Nahata(3/9/2023 6:40 PM IDT)
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Serino Dario(4/9/2023 9:15 AM IDT)
Yasodhar Patnaik(4/9/2023 1:31 PM IDT)
Quentin Higueret(4/9/2023 4:40 PM IDT)
Hakan Summakoğlu(4/9/2023 5:35 PM IDT)
*Guglielmo Sanchini1(4/9/2023 6:46 PM IDT)
Kang Jin Cho(4/9/2023 6:55 PM IDT)
*Alain Michiels(4/9/2023 8:03 PM IDT)
Marco Bandiera(4/9/2023 11:55 PM IDT)
Michael Branicky(5/9/2023 12:35 AM IDT)
*Li Li(5/9/2023 2:56 AM IDT)
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Joaquim Carrapa(5/9/2023 5:00 PM IDT)
Stéphane Higueret(5/9/2023 7:15 PM IDT)
Kamlesh Nahata(5/9/2023 7:49 PM IDT)
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Dieter Beckerle(5/9/2023 8:20 PM IDT)
*David F.H. Dunkley(6/9/2023 12:52 AM IDT)
*Guglielmo Sanchini1(6/9/2023 12:09 PM IDT)
Viswanadh Kandala(6/9/2023 1:14 PM IDT)
Michael Chirozzi(6/9/2023 4:30 PM IDT)
*Dan Dima(6/9/2023 5:38 PM IDT)
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Dheeraj Reddy(9/9/2023 1:46 PM IDT)
Kandala Srinivas(9/9/2023 2:11 PM IDT)
*Guy Daniel Hadas(9/9/2023 5:43 PM IDT)
Clive Tong(9/9/2023 7:37 PM IDT)
Rodrigo Viana Rocha(9/9/2023 8:38 PM IDT)
Sachal Mahajan(9/9/2023 8:51 PM IDT)
*Dan Ismailescu(10/9/2023 4:29 AM IDT)
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Abhinav Bana(11/9/2023 3:17 PM IDT)
*Dario Filatrella(11/9/2023 5:46 PM IDT)
Christoph Schmidt(11/9/2023 6:26 PM IDT)
Sri Mallikarjun J(11/9/2023 7:17 PM IDT)
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*Daniel Copeland & Whitney Fish & Isaac Lawrence(29/9/2023 7:53 PM IDT)
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