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 A067742 Number of middle divisors of n, i.e., divisors in the half-open interval [sqrt(n/2), sqrt(n*2)). 37
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 OFFSET 1,6 COMMENTS From Omar E. Pol, Feb 06 2017: (Start) Conjecture 1: a(n) is also the difference between the number of odd divisors of n less than sqrt(2*n) and the number of odd divisors of n greater than sqrt(2*n). Conjecture 2: Also, alternating row sums of A237048. Conjecture 3: Also central column of A249351. Conjecture 4: The parity of this sequence is A053866 (also A093709, n > 0), the same parity of A000203, A001227 and A000593. Conjecture 5: Indices of odd terms gives A028982. Indices of even terms gives A028983. Conjecture 6: a(n) is also the width of the terrace at the n-th level in the main diagonal of the pyramid described in A245092. Conjecture 7: a(n) is also the number of central subparts of the symmetric representation of sigma(n). For more information see A279387. (End) Conjecture 8: a(n) is also the difference between the number of partitions of n into an odd number of consecutive parts and the number of partitions of n into an even number of consecutive parts. - Omar E. Pol, Feb 24 2017 LINKS R. Zumkeller, Table of n, a(n) for n = 1..10000 Robin Chapman, Kimmo Eriksson and Richard Stanley, On the Number of Divisors of n in a Special Interval: Problem 10847, Amer. Math. Monthly 109, (2002), p. 80. Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015, see page 29 Remarks 6.8(b). [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.] Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016, see Remark 1.3. J. E. Vatne, The sequence of middle divisors is unbounded, arXiv:1607.02122 [math.NT], 2016, shows that there is a subsequence diverging to infinity. FORMULA G.f.: Sum_{k>=1} (-1)^(k-1)*q^binomial(k+1, 2)/(1-q^k). a(A128605(n)) = n and a(m) <> n for m < A128605(n). - Reinhard Zumkeller, Mar 14 2007 It appears that a(n) = A240542(n) - A240542(n-1), the difference between two adjacent Dyck paths as defined in A237593. - Hartmut F. W. Hoft, Feb 06 2017 Conjecture: a(n) = A082647(n) - A131576(n) = A001227(n) - 2*A131576(n). - Omar E. Pol, Feb 06 2017 EXAMPLE a(6)=2 because the 2 middle divisors of 6 (2 and 3) are between sqrt(3) and sqrt(12). MATHEMATICA (* number of middle divisors *) a067742[n_] := Select[Divisors[n], Sqrt[n/2] <= # < Sqrt[2n] &] a067742 (* data *) (* Hartmut F. W. Hoft, Jul 17 2014 *) a[ n_] := If[ n < 1, 0, DivisorSum[ n, 1 &, n/2 <= #^2 < 2 n &]]; (* Michael Somos, Jun 04 2015 *) (* support function a240542[] is defined in A240542 *) b[n_] := a240542[n] - a240542[n-1] Map[b, Range] (* data - Hartmut F. W. Hoft, Feb 06 2017 *) PROG (PARI)  A067742(n) = {sumdiv(n, d, d2 = d^2; n / 2 < d2 && d2 <= n << 1)} \\ M. F. Hasler, May 12 2008 (PARI) a(n) = A067742(n) = {my(d = divisors(n), iu = il = #d \ 2); if(#d <= 2, return(n < 3)); while(d[il]^2 > n>>1, il--); while(d[iu]^2 < (n<<1), iu++); iu - il - 1 + issquare(n/2)} \\ David A. Corneth, Apr 06 2018 CROSSREFS Cf. A067743, A071090 (sums of middle divisors). Cf. A071561 (positions of zeros), A071562 (positions of nonzeros). Relating to Dyck paths: A237593, A240542. Sequence in context: A129561 A259895 A276479 * A302233 A214772 A332036 Adjacent sequences:  A067739 A067740 A067741 * A067743 A067744 A067745 KEYWORD easy,nonn AUTHOR Marc LeBrun, Jan 29 2002 STATUS approved

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Last modified September 24 15:31 EDT 2020. Contains 337321 sequences. (Running on oeis4.)