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

Veikko Pohjola veikko at nordem.fi
Sun Jan 31 13:05:03 CET 2016


Let me return to my original message. A more detailed picture of the way how the triangle behaves is the sequence of the triples {a,b,c} arranged in the increasing order of the smallest side a as follows:
{13, 19, 22}, {17, 22, 31}, {25, 38, 41}, {37, 58, 59}, {41, 58, 71}, {53, 62, 101}, {61, 82, 109}, {65, 79, 122}, {65, 95, 110}, {85, 118, 149}, {89, 121, 158}, {101, 139, 178}, ...
None of the sequences of the individual sides a, b, and c appears in OEIS but I doubt their common interest because they spring from a somewhat artificial-looking geometric construct.
Veikko


Sven Simon kirjoitti 31.1.2016 kello 12.23:

> That should read correctly 2025 = 1³+2³+3³+4³+5³+6³+7³+8³+9³ = (1+2+3+4+5+6+7+8+9)**2
> The text format changed, the exponent was written like normal text after sending, on my screen it was still ok.
> Sven
> 
> -----Ursprüngliche Nachricht-----
> Von: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] Im Auftrag von Sven Simon
> Gesendet: Sonntag, 31. Januar 2016 11:15
> An: 'Sequence Fanatics Discussion list'
> Betreff: [seqfan] Re: Happy 3³+4³+5³+6³+7³+8³+9³
> 
> So 2025 we have
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> 
> 2025 = 1³+2³+3³+4³+5³+6³+7³+8³+9³ = (1+2+3+4+5+6+7+8+9)2
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> 
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> 
> 
> -----Ursprüngliche Nachricht-----
> Von: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] Im Auftrag von Veikko Pohjola
> Gesendet: Samstag, 30. Januar 2016 19:20
> An: Sequence Fanatics Discussion list
> Betreff: [seqfan] Re: Happy 3³+4³+5³+6³+7³+8³+9³
> 
> 
> 
> Dear Giovanni and others,
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> 
> 
> 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, …
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> Happy New Year,
> 
> Veikko Pohjola
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> Giovanni Resta kirjoitti 4.1.2016 kello 10.59:
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>> Other notable properties of 2016:
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>> (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.
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>> (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.
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>> (3) The smallest number equal to the sum of its 31 smallest divisors.
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>> (4) The smallest number whose square root begins with 10 composite digits (so 4, 6, 8 or 9).
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>> (5) If p(n) denotes the n-th prime number, the product of digits of p(p(666)) is 2016.
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>> (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.
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>> (7) The value of y in this geometrical problem  <http://www.numbersaplenty.com/prob2016.png> http://www.numbersaplenty.com/prob2016.png
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>> Giovanni Resta
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