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A082662 Numbers k such that the odd part of k is less than sqrt(2k).

%I A082662 #85 Feb 16 2025 08:32:49
%S A082662 1,2,4,6,8,12,16,20,24,28,32,40,48,56,64,72,80,88,96,104,112,120,128,
%T A082662 144,160,176,192,208,224,240,256,272,288,304,320,336,352,368,384,400,
%U A082662 416,432,448,464,480,496,512,544,576,608,640,672,704,736,768,800
%N A082662 Numbers k such that the odd part of k is less than sqrt(2k).
%C A082662 Theorem: The following eight definitions are equivalent.
%C A082662 (P1) Numbers k such that the odd part of k (A000265(k)) is < sqrt(2k).
%C A082662   (P1) is the new definition, repeated here for convenience. Note that this is not the same as saying A000265(k) < A172471(k), since A172471(k) = floor(sqrt(2*k)).
%C A082662 (P2) Numbers k such that the odd divisors of k are < sqrt(2k).
%C A082662   (P2) and (P1) are obviously equivalent.
%C A082662 (P3) The numbers 1, S_0, S_1, S_2, ..., where
%C A082662   S_m = { 2^(m+1)*(2^m+i) : i = 0 .. 3*2^m - 1 }.
%C A082662 So S_0 = {2,4,6}, S_1 = {8,12,16,20,24,28}, S_2 = {32,40,48,...,120}, S_3 = {128,144,...,496}, ...
%C A082662   The proof that (P3) and (P1) are the same sequence is not difficult and will be added later. (P3) is equivalent to a formula stated without proof (it may have been only an empirical observation) in the original version of this entry.
%C A082662 (P4) Numbers k such that the odd part of k is <= A003056(k).
%C A082662   That is, the odd part of k is <= floor((sqrt(1+8*n)-1)/2). It is more difficult to show this is equivalent to (P1), but it is true.
%C A082662 (P5) Numbers k such that the odd divisors of k are <= A003056(k).
%C A082662   (P5) and (P4) are obviously equivalent.
%C A082662 (P6) Numbers k such that A001227(k) = A082647(k).
%C A082662   (P6) was the original definition. In words, it says that the number of odd divisors of k is equal to the number of ways to write k as a sum of an odd number of consecutive positive integers, or equivalently as a sum of d consecutive positive integers for some d dividing k.  To show that (P6) is equivalent to (P1) one makes use of the Hirschhorn-Hirschhorn article.
%C A082662 (P7) Numbers k such that the odd part of k is <= the sum of divisors of the even part.
%C A082662   (P7) was contributed by _Jaycob Coleman_, Jun 21 2014. To show (P7) is equivalent to (P1), write k as 2^m*s where s is odd.  Equality holds if and only if k is an even perfect number.
%C A082662 (P8) Numbers k such that A000265(k) <= A000203(A006519(k)) or also such that A000265(k) <= A038712(k).
%C A082662   (P8) was contributed by _Michel Marcus_, Aug 14 2014. It is a restatement of (P7).
%C A082662 (End of theorem)
%C A082662 A further equivalent property, (P9), follows at once from (P4). This was conjectured by  _Omar E. Pol_, Apr 18 2017
%C A082662 (P9) These are the numbers k such that the sequence of successive widths in the symmetric representation of sigma(k) is unimodal.
%C A082662 Yet another equivalent property:
%C A082662 (P10) Numbers k >= 1 such if k = i + (i+1) + (i+2) + ... + (i+j-1) for some i >= 1 and j >= 1 then j is odd [Caballero, 2019]. - _Michel Marcus_, Jan 16 2020
%C A082662 This is a subsequence of A005153. - _Jaycob Coleman_, Jun 21 2014
%C A082662 The complement of this sequence is A281005. - _Omar E. Pol_, Apr 18 2017
%C A082662 Subsequence of A174973. - _Omar E. Pol_, Feb 01 2021
%H A082662 Amiram Eldar, <a href="/A082662/b082662.txt">Table of n, a(n) for n = 1..10000</a>
%H A082662 José Manuel Rodríguez Caballero, <a href="http://math.colgate.edu/~integers/t20/t20.Abstract.html">Integers Which Cannot Be Partitioned Into an Even Number of Consecutive Parts</a>, INTEGERS, Volume 19 (2019), #A20.
%H A082662 M. D. Hirschhorn and P. M. Hirschhorn, <a href="http://www.maa.org/sites/default/files/3004420056860.pdf.bannered.pdf">Partitions into Consecutive Parts</a>, Mathematics Magazine: 2003, Volume 76, Number 4, pp. 306-308.
%H A082662 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 38.
%H A082662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EvenPart.html">Even Part</a>
%H A082662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OddPart.html">Odd Part</a>
%F A082662 G.f. = 1 + (1/(1-x)^2) * Sum_{m >= 0} (2^(m+1)*x^(3*2^m-2) * ( x^(3*2^m)*(2^(m+2)*(x-1)-x) - 2^m*(x-1) + x ) ). (This follows from (P3).) :w
%F A082662 - _N. J. A. Sloane_, Feb 02 2021
%F A082662 a(n+1) = a(n) + A053644(A000196(2*a(n))). - _Peter Munn_, Oct 03 2023
%t A082662 cnt[n_] := DivisorSum[n, Boole[OddQ[#] && #>Sqrt[2n]]&]; Select[Range[800], cnt[#]==0&] (* _Jean-François Alcover_, Feb 16 2017 *)
%o A082662 (PARI) isok(n) = my(q = sqrt(2*n)); (sumdiv(n, d, (d%2) && (d < q)) == sumdiv(n, d, d%2)); \\ _Michel Marcus_, Jul 04 2014
%Y A082662 Cf. A281005 (complement).
%Y A082662 Cf. A000196, A000203, A000265, A001227, A003056, A005153, A006519, A038712, A053644, A082647, A116882, A172471, A174973, A237593, A279387.
%K A082662 easy,nonn,nice
%O A082662 1,2
%A A082662 _Naohiro Nomoto_, May 18 2003
%E A082662 Edited by _N. J. A. Sloane_, Jan 28 2021: Replaced original indirect definition by simple direct definition; rearranged comments; provided proofs (not yet included here) that the various definitions are equivalent