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 A066272 Number of anti-divisors of n. 142

%I

%S 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,

%T 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,

%U 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

%N Number of anti-divisors of n.

%C 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.

%C 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.

%C 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.

%C 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.

%C k is an anti-divisor of n if and only if 1 < k < n and | (n mod k) - k/2 | < 1. - _Max Alekseyev_, Jul 21 2007

%C 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. - _Franklin T. Adams-Watters_, Sep 11 2009

%C 2n-1 and 2n+1 are twin primes if and only if n has no odd anti-divisors. (e.g. n=15 has no odd anti-divisors so 29 and 31 are twin primes). - _Jon Perry_, Sep 02 2012

%C Records are in A066464. - _Robert G. Wilson v_, Sep 03 2012

%H Diana Mecum and T. D. Noe, <a href="/A066272/b066272.txt">Table of n, a(n) for n = 1..10000</a>

%H Jon Perry, <a href="/A066272/a066272a.html">The Anti-divisor</a>

%H Jon Perry, <a href="/A066272/a066272.html">The Anti-divisor: Even More Anti-Divisors</a>

%F G.f. sum(k>0, x^(3k) / (1 - x^(2k)) + (x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1))). - _Franklin T. Adams-Watters_, Sep 11 2009

%F a(n) = A000005(2*n-1) + A000005(2*n+1) + A001227(n) - 5. - _Max Alekseyev_, Apr 27 2010

%F a(n) = sum_{i=3..n} (i mod 2) * (3 + floor((2n-1)/i) - ceiling((2n-1)/i) + floor(2n/i) - ceiling(2n/i) + floor((2n+1)/i) - ceiling((2n+1)/i)). - _Wesley Ivan Hurt_, Aug 10 2014

%e 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.

%p antidivisors := proc(n)

%p local a,k;

%p a := {} ;

%p for k from 2 to n-1 do

%p if abs((n mod k)- k/2) < 1 then

%p a := a union {k} ;

%p end if;

%p end do:

%p a ;

%p end proc:

%p A066272 := proc(n)

%p nops(antidivisors(n)) ;

%p end proc:

%p seq(A066272(n),n=1..120); # _R. J. Mathar_, May 24 2010

%t antid[ n_ ] := Select[ Union[ Join[ Select[ Divisors[ 2n - 1 ], OddQ[ # ] && # != 1 & ], Select[ Divisors[ 2n + 1 ], OddQ[ # ] && # != 1 & ], 2n / Select[ Divisors[ 2n ], OddQ[ # ] && # != 1 & ] ] ], # < n & ]; Table[ Length[ antid[ n ] ], {n, 1, 100} ]

%t 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_, Jul 17 2007 *)

%t nd[n_]:=Count[Range[2,n-1],_?(Abs[Mod[n,#]-#/2]<1&)]; Array[nd,110] (* _Harvey P. Dale_, Jul 11 2012 *)

%o (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)))) \\ _Franklin T. Adams-Watters_, Sep 11 2009

%o (PARI) a(n) = if(n>1, numdiv(2*n+1) + numdiv(2*n-1) + numdiv(n/2^valuation(n,2)) - 5, 0) \\ _Max Alekseyev_, Apr 27 2010

%o (PARI) antidivisors(n)=select(t->n%t && t<n, concat(concat(divisors(2*n-1),divisors(2*n+1)), 2*divisors(n)))

%o a(n)=#antidivisors(n) \\ _Charles R Greathouse IV_, May 12 2016

%o (Python)

%o from sympy import divisors

%o def A066272(n):

%o ....return len([d for d in divisors(2*n) if n > d >=2 and n%d]) + len([d for d in divisors(2*n-1) if n > d >=2 and n%d]) + len([d for d in divisors(2*n+1) if n > d >=2 and n%d]) # _Chai Wah Wu_, Aug 11 2014

%Y Cf. A058838, A066464, A066241, A001227, A001511.

%Y See A130799 for the anti-divisors.

%K nonn,easy

%O 1,5

%A _N. J. A. Sloane_, Dec 31 2001

%E More terms from _Robert G. Wilson v_, Jan 02 2002

%E More terms from _Max Alekseyev_, Apr 27 2010

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Last modified November 16 20:13 EST 2019. Contains 329206 sequences. (Running on oeis4.)