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

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, 34, 37, 38, 39, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 101, 103, 106

Offset: 1

Author

Reinhard Zumkeller, Apr 01 2010

Keywords

• nonn

Comments

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

sequences of powers of primes are subsequences;

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

Also the union of A241008 and A241010 (see the link for a proof). - Hartmut F. W. Hoft, Jul 02 2015

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

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Hartmut F. W. Hoft, Proof that this sequence equals union of A241008 and A241010

Crossrefs

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

Cf. A357581.

Programs

Haskell

a174905 n = a174905_list !! (n-1)
a174905_list = filter ((== 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

Maple

filter:= proc(n)
  local d,q;
   d:= numtheory:-divisors(n);
   min(seq(d[i+1]/d[i],i=1..nops(d)-1)) >= 2
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 08 2014

Mathematica

(* it suffices to test adjacent divisors *)
a174905[n_] := Module[{d = Divisors[n]}, ! Apply[Or, Map[2 #[[1]] > #[[2]] &, Transpose[{Drop[d, -1], Drop[d, 1]}]]]]
(* Hartmut F. W. Hoft, Aug 07 2014 *)
Select[Range[106], !MatchQ[Divisors[#], {_, d_, e_, _} /; e < 2d]& ] (* Jean-François Alcover, Jan 31 2018 *)