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A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.

%I A174905 #45 Oct 11 2022 01:00:11
%S A174905 1,2,3,4,5,7,8,9,10,11,13,14,16,17,19,21,22,23,25,26,27,29,31,32,33,
%T A174905 34,37,38,39,41,43,44,46,47,49,50,51,52,53,55,57,58,59,61,62,64,65,67,
%U A174905 68,69,71,73,74,76,79,81,82,83,85,86,87,89,92,93,94,95,97,98,101,103,106
%N A174905 Numbers with no pair (d,e) of divisors such that d < e < 2*d.
%C A174905 A174903(a(n)) = 0; complement of A005279;
%C A174905 sequences of powers of primes are subsequences;
%C A174905 a(n) = A129511(n) for n < 27, A129511(27) = 35 whereas a(27) = 37.
%C A174905 Also the union of A241008 and A241010 (see the link for a proof). - _Hartmut F. W. Hoft_, Jul 02 2015
%C A174905 In other words: numbers n with the property that all parts in the symmetric representation of sigma(n) have width 1. - _Omar E. Pol_, Dec 08 2016
%C A174905 Sequence A357581 shows the numbers organized in columns of a square array by the number of parts in their symmetric representation of sigma. - _Hartmut F. W. Hoft_, Oct 04 2022
%H A174905 Reinhard Zumkeller, <a href="/A174905/b174905.txt">Table of n, a(n) for n = 1..10000</a>
%H A174905 Hartmut F. W. Hoft, <a href="/A174905/a174905.pdf">Proof that this sequence equals union of A241008 and A241010</a>
%p A174905 filter:= proc(n)
%p A174905   local d,q;
%p A174905    d:= numtheory:-divisors(n);
%p A174905    min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
%p A174905 end proc:
%p A174905 select(filter, [$1..1000]); # _Robert Israel_, Aug 08 2014
%t A174905 (* it suffices to test adjacent divisors *)
%t A174905 a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
%t A174905 (* _Hartmut F. W. Hoft_, Aug 07 2014 *)
%t A174905 Select[Range[106], !MatchQ[Divisors[#], {___, d_, e_, ___} /; e < 2d]& ] (* _Jean-François Alcover_, Jan 31 2018 *)
%o A174905 (Haskell)
%o A174905 a174905 n = a174905_list !! (n-1)
%o A174905 a174905_list = filter ((== 0) . a174903) [1..]
%o A174905 -- _Reinhard Zumkeller_, Sep 29 2014
%Y A174905 Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.
%Y A174905 Cf. A357581.
%K A174905 nonn
%O A174905 1,2
%A A174905 _Reinhard Zumkeller_, Apr 01 2010