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A066272
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Number of anti-divisors of n.
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105
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0, 0, 1, 1, 2, 1, 3, 2, 2, 3, 3, 2, 4, 3, 3, 2, 5, 4, 3, 3, 3, 5, 5, 2, 5, 3, 5, 5, 3, 3, 5, 6, 5, 3, 5, 2, 5, 7, 5, 4, 4, 5, 5, 3, 7, 5, 5, 3, 6, 6, 3, 7, 7, 3, 5, 3, 5, 7, 7, 6, 4, 5, 7, 2, 5, 5, 9, 7, 3, 5, 5, 6, 7, 5, 5, 5, 9, 5, 3, 5, 6, 7, 7, 4, 8, 5, 7, 7, 3, 5, 5, 5, 7, 9, 9, 1, 7, 8, 5, 4, 5, 7, 7, 7, 9
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Anti-divisors are the numbers that do not divide a number by the largest possible margin. E.g. 20 has anti-divisors 3, 8 and 13. An alternative name for anti-divisor is unbiased non-divisors.
Definition: If an odd number i in the range 1 < i <= n divides N where N is any one of 2n-1, 2n or 2n+1 then d = N/i is called an anti-divisor of n. The numbers 1 and 2 have no anti-divisors.
Equivalently, an anti-divisor of n is a number d in the range [2,n-1] which does not divide n and is either a (necessarily odd) divisor of 2n-1 or 2n+1, or a (necessarily even) divisor of 2n.
Thus an anti-divisor of n is an integer d in [2,n-1] such that n == (d-1)/2, d/2, or (d+1)/2 (mod d), the class of d being -1, 0, or 1, respectively.
Comment from Max Alekseyev, Jul 21 2007: k is a non-divisor of n iff 1 < k < n and | (n mod k) - k/2 | < 1.
The number of even anti-divisors of n is one less than the number of odd divisors of n; specifically, all but the largest odd divisor multiplied by the power of two dividing 2n (i.e., 2^A001151(n)). For example, the odd divisors of 18 are 1, 3, and 9, so the even anti-divisors of 18 are 1*4 = 4 and 3*4 = 12. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
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LINKS
| Diana Mecum and T. D. Noe, Table of n, a(n) for n=1..10000
Jon Perry, Anti-divisors [Broken link]
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]
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FORMULA
| G.f. sum(k>0, x^(3k) / (1 - x^(2k)) + (x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1))). [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
a(n) = A000005(2*n-1) + A000005(2*n+1) + A001227(n) - 5 [From Max Alekseyev (maxale(AT)gmail.com), Apr 27 2010]
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EXAMPLE
| For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 18 are 4, 5, 7, 12. Therefore a(18) = 4.
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MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 24 2010: (Start)
antidivisors := proc(n) local a, k; a := {} ; for k from 2 to n-1 do if abs((n mod k)- k/2) < 1 then a := a union {k} ; end if; end do: a ; end proc:
A066272 := proc(n) nops(antidivisors(n)) ; end proc: seq(A066272(n), n=1..120) ; (End)
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MATHEMATICA
| antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2*n ], OddQ[ # ] && # != 1 & ] ] ] }, # < n & ] ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]
f[n_] := Length@ Complement[ Sort@Join[ Select[ Union@ Flatten@ Divisors[{2 n - 1, 2 n + 1}], OddQ@ # && # < n &], Select[ Divisors[2 n], EvenQ@ # && # < n &]], Divisors@ n]; Array[f, 105] (* Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 17 2007 *) ...........
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PROG
| (PARI) al(n)=Vec(sum(k=1, n, (x^(3*k)+x*O(x^n))/(1-x^(2*k))+(x^(3*k+1)+x^(3*k+2)+x*O(x^n))/(1-x^(2*k+1)))) [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
(PARI) { a(n) = if(n>1, numdiv(2*n+1) + numdiv(2*n-1) + numdiv(n/2^valuation(n, 2)) - 5, 0) } [From Max Alekseyev (maxale(AT)gmail.com), Apr 27 2010]
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CROSSREFS
| Cf. A058838, A066464, A066241. A130799 gives the actual anti-divisors.
Cf. A001227, A001511. [From Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Sep 11 2009]
Sequence in context: A023135 A191654 A205784 * A058773 A122805 A103981
Adjacent sequences: A066269 A066270 A066271 * A066273 A066274 A066275
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 31, 2001
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Jan 02 2002
More terms from Max Alekseyev (maxale(AT)gmail.com), Apr 27 2010
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