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A212308
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Numbers with no proper divisor that is not in an arithmetic progression of at least three proper divisors.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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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}.
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MAPLE
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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:
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MATHEMATICA
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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;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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