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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 (list; graph; refs; listen; history; text; internal format)
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 t|s*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), 685--692 MR10,105b; Zbl 32,13 (see Theorem 3).

Paul Erdős, Some unconventional problems in number theory, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 73-82.

Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.

Paul Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692-696, 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 !! (n-1)

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[i-1], 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

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Last modified December 3 08:35 EST 2022. Contains 358515 sequences. (Running on oeis4.)