%I M4093 #109 Dec 31 2023 00:57:08
%S 6,12,15,18,20,24,28,30,35,36,40,42,45,48,54,56,60,63,66,70,72,75,77,
%T 78,80,84,88,90,91,96,99,100,102,104,105,108,110,112,114,117,120,126,
%U 130,132,135,138,140,143,144,150,153,154,156,160,162,165,168,170,174,175,176
%N Numbers having divisors d,e with d < e < 2*d.
%C The arithmetic and harmonic means of A046793(n) and a(n) are both integers.
%C n is in this sequence iff n is a multiple of some term in A020886.
%C 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
%C A174903(a(n)) > 0; complement of A174905. - _Reinhard Zumkeller_, Apr 01 2010
%C Also numbers n such that A239657(n) > 0. - _Omar E. Pol_, Mar 23 2014
%C 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
%C 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
%C 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
%C 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
%C 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
%C From _Bob Andriesse_, Nov 26 2023: (Start)
%C 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.
%C 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.
%C 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)
%D R. K. Guy, Unsolved Problems in Number Theory, E3.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A005279/b005279.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1948-06.pdf">On the density of some sequences of numbers</a>, Bull. Amer. Math. Soc. 54 (1948), 685--692 MR10,105b; Zbl 32,13 (see Theorem 3).
%H Paul Erdős, <a href="http://www.numdam.org/item/?id=AST_1979__61__73_0">Some unconventional problems in number theory</a>, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 73-82.
%H Paul Erdős, <a href="https://www.jstor.org/stable/2689842">Some unconventional problems in number theory</a>, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
%H Paul Erdős, <a href="https://doi.org/10.1112/jlms/s1-39.1.692">On some applications of probability to analysis and number theory</a>, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692-696, <a href="https://users.renyi.hu/~p_erdos/1964-15.pdf">alternative link</a>.
%H Planet Math, <a href="http://planetmath.org/IntegerContraharmonicMeans">Integer Contraharmonic Means</a>, Proposition 4.
%H Planet Math, <a href="http://planetmath.org/ContraharmonicProportion">Contraharmonic proportion</a>
%H Robert G. Wilson v, <a href="/A005279/a005279.pdf">Letter, N.D.</a>
%F a(n) = A010814(n)/2. - _Omar E. Pol_, Dec 04 2016
%p 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
%t aQ[n_] := Select[Partition[Divisors[n], 2, 1], #[[2]] < 2 #[[1]] &] != {}; Select[Range[178], aQ] (* _Jayanta Basu_, Jun 28 2013 *)
%o (Haskell)
%o a005279 n = a005279_list !! (n-1)
%o a005279_list = filter ((> 0) . a174903) [1..]
%o -- _Reinhard Zumkeller_, Sep 29 2014
%o (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
%Y Subsequence of A024619 and hence of A002808.
%Y Cf. A010814, A089341, A020886, A046793, A174903, A174905, A237271, A237593, A239657.
%K nonn
%O 1,1
%A _N. J. A. Sloane_, _Robert G. Wilson v_