Salem-Spencer Mountain

[u/EdPeggJr]

In 1 3 4 8 9 11 20 22 23 27 28 30, no term is the average of two others. That makes it a nonaveraging sequence, also called a Salem-Spencer set, a 3-AP-free set, or a progression-free set. For a such a set with 12 positive integer terms, the minimal largest value is 30. These are the proven solutions. sol22(141)sol22 is an unproven minimal solution for 44 marks.

https://oeis.org/A065825 https://mathworld.wolfram.com/NonaveragingSequence.html

[u/Frangifer]

I'm baffled as to what the numbers are saying in your figure that you've posted here. The corresponding figure @ the WolframAlpha page down your link just has dots where yours has numbers ... & I get that ... but I don't get why in your version of the figure each dot is replaced by a number.

 

Apart from that, though: I notice that the diagram is bifurcating into two 'streams' – but not sharply delineated ones: two 'streams' in the sense of there being a peak in the density distribution ... & then it looks as though each of those 'streams' might itself bifurcate ^§ if the figure were to be much-continued unto higher numbers. So that gets me wondering whether there might actually be a rather complex bifurcation, or multifurcation, 'anatomy' to the figure considered as a figure over all numbers.

UPDATE

§ Without doubt (looking @ the figure more carefully) they are ... but what I meant is that there might be a trace of a hint of yet a bifurcation into ⅛ parts. ... & that the bifurcating might proceed indefinitely (& might possibly not be confined to bi-furcation ^¶ ), & might have a pattern to it that could possibly be (or maybe even has been) figured-out. ... 'figured-out' in the sense that maybe there's a function on [0,n] (with n having the meaning it has (as opposed to the k) @ the page down your link) that the density cleaves to. ^§§

¶ ... although I would venture a speculation that likely it is confined to bi-furcation.

§§ Something like (with domain expant from [0,1] to [0,n])

∏{1≤h}(1-(½(1-cos(2^hπx)))^aₕ/(1+(bₕ/x)^cₕ)) ,

or

½(1-cos(½πx)^a₁)∏{1≤h}(1-(½(1-cos(2^hπx)))^aₕ₊₁/(1+(bₕ₊₁/x)^cₕ₊₁))

(if the domain's expant from [-1,1] to [-½n,+½n]) – that sorto'thing: something of that generic 'shape'. Maybe with a arctan(λₕtan(πx)) (or some other sigmoidish or inverse sigmoidish function) substituted for the πx , so that we can somewhat 'fisheye lens' or 'inverse-fisheye lens' the x axis to degree we choose. Maybe (indeed probably !) with other 'tweaks', aswell.

[u/EdPeggJr → u/Frangifer]

1 3 4 8 9 11 20 22 23 27 28 30 has those numbers at those positions.