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. Here are the known minimal sets (A065825).
SalemSpencer=
;
And here’s a visualization of the same data as Salem-Spencer Mountain:
Out[]=
We can check the values.
In[]:=
AllTrue[Subsets[#,{3}],#[[1]]+#[[3]]!=2#[[2]]&]&/@SalemSpencer
Out[]=
{True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True}
Candidates for 44 and 45 are not yet proven minimal.
In[]:=
candidate44=Join[SalemSpencer[[22]],SalemSpencer[[22]]+141]
Out[]=
{1,2,7,9,10,14,20,22,23,25,29,46,50,52,53,55,61,65,66,68,73,74,142,143,148,150,151,155,161,163,164,166,170,187,191,193,194,196,202,206,207,209,214,215}
candidate45={1,2,6,8,12,17,19,20,24,25,27,43,45,51,54,55,58,60,64,72,76,79,129,145,147,154,155,159,160,167,169,170,172,176,201,202,206,208,212,217,219,220,224,225,227};

CITE THIS NOTEBOOK

Salem-Spencer mountain: nonaveraging sequence​
by Ed Pegg​
Wolfram Community, STAFF PICKS, July 31, 2026
​https://community.wolfram.com/groups/-/m/t/3770853