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A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.

%I A212308 #18 Sep 26 2020 11:43:03
%S A212308 1,6,12,15,18,24,30,36,45,48,54,60,66,72,75,84,90,91,96,108,120,132,
%T A212308 135,144,150,162,168,180,192,198,216,225,240,252,264,270,276,288,300,
%U A212308 306,312,324,330,336,360,375,384,396,405,420,432,435,450,480,486,504
%N A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.
%C A212308 Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.
%C A212308 Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - _Robert Israel_, Apr 13 2020
%H A212308 Robert Israel, <a href="/A212308/b212308.txt">Table of n, a(n) for n = 1..10000</a>
%e A212308 36 appears in this sequence because its proper divisors are 1, 2, 3, 4, 6, 9, 12 and 18, each of which appears in at least one of the following arithmetic progressions of at least three proper divisors of 36: {1, 2, 3, 4}, {3, 6, 9, 12}, {6, 12, 18}.
%p A212308 filter:= proc(n) local S,D,tau,a,b;
%p A212308   S:= numtheory:-divisors(n) minus {n};
%p A212308   D:= sort(convert(S,list));
%p A212308   tau:= nops(D);
%p A212308   for a from 1 to tau-2 do for b from a+1 to tau-1 do
%p A212308     if member(2*D[b]-D[a],D) then
%p A212308       S:= S minus {D[a],D[b],2*D[b]-D[a]};
%p A212308       if S = {} then return true fi;
%p A212308     fi
%p A212308   od od;
%p A212308   false;
%p A212308 end proc:
%p A212308 filter(1):= true:
%p A212308 select(filter, [$1..1000]); # _Robert Israel_, Apr 13 2020
%t A212308 filterQ[n_] := Module[{S, D, tau, a, b}, S = Most @ Divisors[n]; D = S; tau = Length[D]; For[a = 1, a <= tau - 2, a++, For[b = a + 1, b <= tau - 1, b++, If [MemberQ[D, 2 D[[b]] - D[[a]]], S = S ~Complement~ {D[[a]], D[[b]], 2 D[[b]] - D[[a]]}; If[S == {}, Return[True]]]]]; False];
%t A212308 filterQ[1] = True;
%t A212308 Select[Range[1000], filterQ] (* _Jean-François Alcover_, Sep 26 2020, after _Robert Israel_ *)
%Y A212308 Cf. A091010, A091011.
%Y A212308 Contains A033845, A129521.
%K A212308 nonn
%O A212308 1,2
%A A212308 _William Rex Marshall_, Oct 24 2013