

A005279


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


31



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

The arithmetic and harmonic means of A046793(n) and a(n) are 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 (1948) shows that this sequence has a natural density, so a(n) ~ k*n 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 k such that at least one of the parts in the symmetric representation of sigma(k) has width > 1.  Omar E. Pol, Dec 08 2016
Erdős conjectured that the asymptotic density of this sequence is 1. The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 32, 392, 4312, 45738, 476153, 4911730, 50359766, 513682915, 5224035310, ...  Amiram Eldar, Jul 21 2020
Numbers with at least one partition into two distinct parts (s,t), s<t, such that ts*n.  Wesley Ivan Hurt, Jan 16 2022


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
Paul Erdős, On the density of some sequences of numbers, Bull. Amer. Math. Soc. 54 (1948), 685692 MR10,105b; Zbl 32,13 (see Theorem 3).
Paul Erdős, Some unconventional problems in number theory, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 7382.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 6770.
Paul Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692696, alternative link.
Planet Math., Integer Contraharmonic Means, Proposition 4.
Planet Math., Contraharmonic proportion
Robert 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: A219095 A107487 A092671 * A343281 A129512 A259366
Adjacent sequences: A005276 A005277 A005278 * A005280 A005281 A005282


KEYWORD

nonn


AUTHOR

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


STATUS

approved



