Finding counterexample

Number Theory Level 5

Let a 1 = 1 a_1=1 and a n + 1 = 1 + a 1 2 + a 2 2 + ⋯ + a n 2 n a_{n+1}=\dfrac{1+a_1^2+a_2^2+\cdots+a_n^2}{n} .

Find the smallest integer n n such that a n a_n is not an integer.


The answer is 44.

Khang Nguyen Thanh by Khang Nguyen Thanh , 27, Vietnam

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2 solutions

This is known as Göbel's Sequence . Note that the sequence in the link is offset from this one by 1 1 .

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Can you have any other solution ????

Kushal Bose - 4 years, 11 months ago

If this can't be shown then rather than the question this must be a note of information

Aakash Khandelwal - 4 years, 11 months ago

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That's a good idea. If one didn't know about Göbel's Sequence then it can't be expected for them to be able to solve the problem on their own. I posted the link just to provide more information to others about this special sequence.

Brian Charlesworth - 4 years, 11 months ago

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interesting problem.

Srikanth Tupurani - 1 year, 9 months ago

@Jon Haussmann -Did you know about the sequence or did you find a way? I'd be curious to see a solution to this without using the sequence.

Sal Gard - 4 years, 10 months ago

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Yes, I already knew about the sequence. There are a handful of good examples of patterns that seem continue for some time, but then stop after a while. This is one of them.

Jon Haussmann - 4 years, 10 months ago

Here is how far I could go ..

an+1 = (1 + a1^2 + ... (an-1)^2 + an^2 ) / n = (1 + a1^2 + ... (an-1)^2 ) / n + (an^2 ) / n = ((1 + a1^2 + ... (an-1)^2 ) / (n -1 )) * n/(n-1) + (an^2 ) / n = ((1 + a1^2 + ... (an-1)^2 ) / (n -1 )) * (n-1)/n + (an^2 ) / n = an * (n-1)/n + (an^2 ) / n

Nitin Gupta - 4 years, 6 months ago
Bob Kadylo
Nov 12, 2016

By just pencil and paper calculation, I determined that the sequence began: [1, 2, 3, 5, 10, 28, 154, 3520, 1551880,...] and grew VERY rapidly

Searching OEIS with the first eight terms of the sequence was sufficient for Sloane's phenomenal resource to find A003504. I had never heard of Göbel's Sequence before but because of this interesting problem - the discovery happened and further explorations ensued.

It turns out that the 44th term is the first nonintegral value in the sequence. The 43rd term has approximately 89,288,343,500 decimal digits !!

Answer= 44 \boxed {44}

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