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

1,2

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

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 *)

Prog

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

Crossrefs

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

Cf. A357581.

Sequence in context: A116358 A060683 A129511 * A260402 A368087 A191849

Adjacent sequences: A174902 A174903 A174904 * A174906 A174907 A174908

Keyword

nonn

Author

Reinhard Zumkeller, Apr 01 2010

Status

approved