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

[Frank Adams-Watters]

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Franklin T. Adams-Watters


-----Original Message-----
From: Veikko Pohjola <veikko at nordem.fi>
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Sent: Sun, Jan 31, 2016 11:58 am
Subject: [seqfan] Re: Happy 3³+4³+5³+6³+7³+8³+9³

Sorry,I need to make a correction. The triple {65, 95, 110} is a multiple of the first. The idea was to eliminate both the multiples and the mirror images. Thus, the sequence of the smallest side is 13, 17, 25, 37, 41, 53, 61, 65, 85, 89, 101, …VeikkoVeikko Pohjola kirjoitti 31.1.2016 kello 14.05:> 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>> >> >> >> 2025 = 1³+2³+3³+4³+5³+6³+7³+8³+9³ = (1+2+3+4+5+6+7+8+9)2>> >> >> >> >> >> >> >> -----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,>> >> >> >> 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> http://www.numbersaplenty.com/prob2016.png>> >>> >> >>> Giovanni Resta>> >>> >> >>> >> >>> _______________________________________________>> >>> >> >>> Seqfan Mailing list -  <http://list.seqfan.eu/> http://list.seqfan.eu/>> >> >> >> >> >> _______________________________________________>> >> >> >> Seqfan Mailing list -  <http://list.seqfan.eu/> http://list.seqfan.eu/>> >> >> _______________________________________________>> >> Seqfan Mailing list - http://list.seqfan.eu/>> >> >> _______________________________________________>> >> Seqfan Mailing list - http://list.seqfan.eu/> > > _______________________________________________> > Seqfan Mailing list - http://list.seqfan.eu/_______________________________________________Seqfan Mailing list - http://list.seqfan.eu/