

A005279


Numbers having divisors d,e with d < e < 2d.
(Formerly M4093)


25



6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 143, 144, 150, 153, 154, 156, 160, 162, 165, 168, 170, 174, 175, 176
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OFFSET

1,1


COMMENTS

Arithmetic and harmonic means of A046793(n) and a(n) both integers.
n is in this sequence iff n is a multiple of some term in A020886.
a(n) is also a positive integer v for which there exists a smaller positive integer u such that the contraharmonic mean (uu+vv)/(u+v) is an integer c (in fact, there are two distinct values u giving with v the same c).  Pahikkala Jussi, Dec 14 2008
A174903(a(n)) > 0; complement of A174905.  Reinhard Zumkeller, Apr 01 2010
Also numbers n such that A239657(n) > 0.  Omar E. Pol, Mar 23 2014
Erdős shows that this sequence has a natural density, so a(n) ~ kn for some constant k. It can be shown that k < 3.03, and by numerical experiments it seems that k is around 1.8.  Charles R Greathouse IV, Apr 22 2015
Numbers n with the property that at least one of the parts in the symmetric representation of sigma(n) has width > 1.  Omar E. Pol, Dec 08 2016


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, E3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

T. D. Noe and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000, first 1000 terms from T. D. Noe
P. Erdõs, On the density of some sequences of numbers, Bull. Amer. Math. Soc. 54 (1948), 685692 MR10,105b; Zbl 32,13 (see Theorem 3).
Planet Math., Proposition 4
Planet Math., Definition
R. G. Wilson, V, Letter, N.D.


FORMULA

a(n) = A010814(n)/2.  Omar E. Pol, Dec 04 2016


MAPLE

isA005279 := proc(n) local divs, d, e ; divs := numtheory[divisors](n) ; for d from 1 to nops(divs)1 do for e from d+1 to nops(divs) do if divs[e] < 2*divs[d] then RETURN(true) ; fi ; od: od: RETURN(false) : end; for n from 3 to 300 do if isA005279(n) then printf("%d, ", n) ; fi ; od : # R. J. Mathar, Jun 08 2006


MATHEMATICA

aQ[n_] := Select[Partition[Divisors[n], 2, 1], #[[2]] < 2 #[[1]] &] != {}; Select[Range[178], aQ] (* Jayanta Basu, Jun 28 2013 *)


PROG

(Haskell)
a005279 n = a005279_list !! (n1)
a005279_list = filter ((> 0) . a174903) [1..]
 Reinhard Zumkeller, Sep 29 2014
(PARI) is(n)=my(d=divisors(n)); for(i=3, #d, if(d[i]<2*d[i1], return(1))); 0 \\ Charles R Greathouse IV, Apr 22 2015


CROSSREFS

Cf. A010814, A089341, A020886, A046793, A174903, A174905, A237271, A237593, A239657.
Sequence in context: A107487 A092671 A239658 * A129512 A259366 A196391
Adjacent sequences: A005276 A005277 A005278 * A005280 A005281 A005282


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Robert G. Wilson v


STATUS

approved



