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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

To make the series you start at zero. Then you add one, and one, and two and two, and three, and 3, and 4, and 4, and 5 and 5 and 6 and 6 and 7 and 7, and continue with two of each increasing natural number.

Here is the beginning of the series.

0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90...

Try to find something interesting about these numbers and say what it is.

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

...100, 110, 121, 132, 144, 156, 169, 182, 196, 210...

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

...225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420...

**igloo** **myrtilles** **fourmis**

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

...441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702...

**igloo** **myrtilles** **fourmis**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,748

Hi John;

...441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702...

Next one is 729 and the after that is 756, isn't that interesting enough!

Here are 2 ways to generate that sequence:

with a1 = 0 , a2 = 1, ...

Except for 2 there is not a single prime!

**In mathematics, you don't understand things. You just get used to them.I have the result, but I do not yet know how to get it.All physicists, and a good many quite respectable mathematicians are contemptuous about proof.**

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

This sequence can also be written as:

1x0, 1x1, 2x1, 2x2, 3x2, 3x3, 4x3, 4x4, ...

So you go from term to term by adding one to alternating sides of the product.

Why did the vector cross the road?

It wanted to be normal.

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,562

Yes, you guys are smarter than me.

Mathsy had my same answer, but bobby did stuff I didn't know.

Good Job!

**igloo** **myrtilles** **fourmis**

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 84,748

Matthew Vandermast wrote:

Also, interesting is the fact that this sequence is the maximum product of two integers whose sum is n.

For instance:

0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90

a19 = 90 so no sum of 2 numbers that equals 19 can have a product bigger than 90.

Example:

18 + 1 =19 and 18*1<= 90

17 + 2 =19 and 17*2<= 90

16 + 3 =19 and 16*3<= 90

15 + 4 =19 and 15*4<= 90

14 + 5 =19 and 14*5<= 90

13 + 6 =19 and 13*6<= 90

12 + 7 =19 and 12*7<= 90

11 + 8 =19 and 11*8<= 90

10 + 9 =19 and 10*9<= 90

**In mathematics, you don't understand things. You just get used to them.I have the result, but I do not yet know how to get it.All physicists, and a good many quite respectable mathematicians are contemptuous about proof.**

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