[seqfan] Re: Happy 3³+4³+5³+6³+7³+8³+9³

[Veikko Pohjola]

Dear Giovanni and others,

The French have a custom to send New Year's greetings at the end of January. So I got the idea to play a little with the charming geometrical problem, number 7 below. The side lengths of the triangles, where all the sides are integral and satisfy the condition of green squares x+y=z, form each a nice sequence as the triangle grows. The smallest side goes like this: 13,17,25,37,41,53,61, …

Happy New Year,
Veikko Pohjola

Giovanni Resta kirjoitti 4.1.2016 kello 10.59:

> Other notable properties of 2016:
> 
> (1) The middle value y of the smallest amicable triple, i.e., 3 numbers x < y < z such that sigma(x)=sigma(y)=sigma(z) = x+y+z \sigma(n) is the sum of the divisors of n.
> 
> (2) The only number n such that n^3 + n^2 is strictly pandigital, i.e., it contains all the digits 0 to 9 exactly once.
> 
> (3) The smallest number equal to the sum of its 31 smallest divisors.
> 
> (4) The smallest number whose square root begins with 10 composite digits (so 4, 6, 8 or 9).
> 
> (5) If p(n) denotes the n-th prime number, the product of digits of p(p(666)) is 2016.
> 
> (6) T(rev(iT(2016))) = 666, where T(n) is the n-th triangular number, rev(n) is the reverse of n, and iT(T(n))=n.
> 
> (7) The value of y in this geometrical problem http://www.numbersaplenty.com/prob2016.png
> 
> Giovanni Resta
> 
> 
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