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A005279 Numbers having divisors d,e with d < e < 2*d.
(Formerly M4093)
33
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
Appears to be the set of numbers x such that there exist numbers y and z satisfying the condition (x^2+y^2)/(x^2+z^2) = (x+y)/(x+z). For example, (15^2+10^2)/(15^2+3^2) = (15+10)/(15+3), so 15 is in the sequence. - Gary Detlefs, Apr 01 2023
From Bob Andriesse, Nov 26 2023: (Start)
Rewriting (x^2+y^2) / (x^2+z^2) = (x+y) / (x+z) as (x^2+y^2) / (x+y) = (x^2+z^2) / (x+z) has the advantage that the values on both sides of the = sign in the given example become integers. A possible sequence with the name: "k's for which r = (k^2+m^2) / (k+m) can be an integer while m<k" appears to have the same terms as this sequence, with the corresponding m's being A053629(n) and the r's being A009003(n). If (k^2+m^2) / (k+m) = r and m satisfies the divisibility condition, then r-m also does, because (k^2 + (r-m)^2) / (k + (r-m)) = r as well, confirming Pahikkala Jussi's comment about the existence of two distinct values for his u.
The fact that 15 is in the sequence is not so much because (15^2 + 10^2) / (15^2 + 3^2) = 1.3888... = (15+10) / (15+3), as indicated by Gary Detlefs, but rather because (15+10) | (15^2 + 10^2). And since r = (15^2 + 10^2) / (15+10) = 13, the second value that satisfies the divisibility condition is 13-10 = 3, so (15^2 + 3^2) / (15+3) = 13 as well.
Since (k+m)| (k^2 + m^2) is equivalent to (k+m) | 2*k^2 as well as to (k+m) | 2*m^2, both of these alternative divisibility conditions can be used to generate the same sequence too. (End)
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
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.
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
Subsequence of A024619 and hence of A002808.
Sequence in context: A219095 A107487 A092671 * A343281 A129512 A259366
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)