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A082662
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Numbers k such that the odd part of k is less than sqrt(2k).
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12
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1, 2, 4, 6, 8, 12, 16, 20, 24, 28, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120, 128, 144, 160, 176, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800
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OFFSET
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1,2
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COMMENTS
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Theorem: The following eight definitions are equivalent.
(P1) Numbers k such that the odd part of k (A000265(k)) is < sqrt(2k).
(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)).
(P2) Numbers k such that the odd divisors of k are < sqrt(2k).
(P2) and (P1) are obviously equivalent.
(P3) The numbers 1, S_0, S_1, S_2, ..., where
S_m = { 2^(m+1)*(2^m+i) : i = 0 .. 3*2^m - 1 }.
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}, ...
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.
(P4) Numbers k such that the odd part of k is <= A003056(k).
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.
(P5) Numbers k such that the odd divisors of k are <= A003056(k).
(P5) and (P4) are obviously equivalent.
(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.
(P7) Numbers k such that the odd part of k is <= the sum of divisors of the even part.
(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.
(P8) was contributed by Michel Marcus, Aug 14 2014. It is a restatement of (P7).
(End of theorem)
A further equivalent property, (P9), follows at once from (P4). This was conjectured by Omar E. Pol, Apr 18 2017
(P9) These are the numbers k such that the sequence of successive widths in the symmetric representation of sigma(k) is unimodal.
Yet another equivalent property:
(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
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LINKS
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Eric Weisstein's World of Mathematics, Even Part
Eric Weisstein's World of Mathematics, Odd Part
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FORMULA
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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
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MATHEMATICA
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cnt[n_] := DivisorSum[n, Boole[OddQ[#] && #>Sqrt[2n]]&]; Select[Range[800], cnt[#]==0&] (* Jean-François Alcover, Feb 16 2017 *)
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PROG
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(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
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CROSSREFS
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Cf. A000196, A000203, A000265, A001227, A003056, A005153, A006519, A038712, A053644, A082647, A116882, A172471, A174973, A237593, A279387.
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KEYWORD
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easy,nonn,nice
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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