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A212308 Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors. 1
1, 6, 12, 15, 18, 24, 30, 36, 45, 48, 54, 60, 66, 72, 75, 84, 90, 91, 96, 108, 120, 132, 135, 144, 150, 162, 168, 180, 192, 198, 216, 225, 240, 252, 264, 270, 276, 288, 300, 306, 312, 324, 330, 336, 360, 375, 384, 396, 405, 420, 432, 435, 450, 480, 486, 504 (list; graph; refs; listen; history; text; internal format)
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
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
Contains A033845, A129521.
Sequence in context: A356736 A066312 A309944 * A089341 A256617 A252044
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)