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Numbers with no pair (d,e) of divisors such that d < e < 2*d.
17

%I #45 Oct 11 2022 01:00:11

%S 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 34,37,38,39,41,43,44,46,47,49,50,51,52,53,55,57,58,59,61,62,64,65,67,

%U 68,69,71,73,74,76,79,81,82,83,85,86,87,89,92,93,94,95,97,98,101,103,106

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

%C A174903(a(n)) = 0; complement of A005279;

%C sequences of powers of primes are subsequences;

%C a(n) = A129511(n) for n < 27, A129511(27) = 35 whereas a(27) = 37.

%C Also the union of A241008 and A241010 (see the link for a proof). - _Hartmut F. W. Hoft_, Jul 02 2015

%C 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 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 Reinhard Zumkeller, <a href="/A174905/b174905.txt">Table of n, a(n) for n = 1..10000</a>

%H Hartmut F. W. Hoft, <a href="/A174905/a174905.pdf">Proof that this sequence equals union of A241008 and A241010</a>

%p filter:= proc(n)

%p local d,q;

%p d:= numtheory:-divisors(n);

%p min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Aug 08 2014

%t (* it suffices to test adjacent divisors *)

%t a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]

%t (* _Hartmut F. W. Hoft_, Aug 07 2014 *)

%t Select[Range[106], !MatchQ[Divisors[#], {___, d_, e_, ___} /; e < 2d]& ] (* _Jean-François Alcover_, Jan 31 2018 *)

%o (Haskell)

%o a174905 n = a174905_list !! (n-1)

%o a174905_list = filter ((== 0) . a174903) [1..]

%o -- _Reinhard Zumkeller_, Sep 29 2014

%Y Cf. A000040, A000961, A001248, A005279, A030078, A030514, A129511, A174903, A237271, A237593, A241008, A241010.

%Y Cf. A357581.

%K nonn

%O 1,2

%A _Reinhard Zumkeller_, Apr 01 2010