OFFSET
1,2
COMMENTS
Equivalently, the numbers with exactly one divisor that is not in an arithmetic progression of at least three divisors.
Contains p^j*(2*p-1)^k for j,k>=1 if p and 2*p-1 are primes. - Robert Israel, Apr 13 2020
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
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}.
MAPLE
filter:= proc(n) local S, D, tau, a, b;
S:= numtheory:-divisors(n) minus {n};
D:= sort(convert(S, list));
tau:= nops(D);
for a from 1 to tau-2 do for b from a+1 to tau-1 do
if member(2*D[b]-D[a], D) then
S:= S minus {D[a], D[b], 2*D[b]-D[a]};
if S = {} then return true fi;
fi
od od;
false;
end proc:
filter(1):= true:
select(filter, [$1..1000]); # Robert Israel, Apr 13 2020
MATHEMATICA
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];
filterQ[1] = True;
Select[Range[1000], filterQ] (* Jean-François Alcover, Sep 26 2020, after Robert Israel *)
CROSSREFS
KEYWORD
nonn
AUTHOR
William Rex Marshall, Oct 24 2013
STATUS
approved