This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A066272 Number of anti-divisors of n. 141
 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; text; internal format)
 OFFSET 1,5 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. 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 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 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 Records are in A066464. - Robert G. Wilson v, Sep 03 2012 LINKS Diana Mecum and T. D. Noe, Table of n, a(n) for n = 1..10000 Jon Perry, The Anti-divisor Jon Perry, The Anti-divisor: Even More Anti-Divisors FORMULA 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 a(n) = A000005(2*n-1) + A000005(2*n+1) + A001227(n) - 5. - Max Alekseyev, Apr 27 2010 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 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. MAPLE 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); # R. J. Mathar, May 24 2010 MATHEMATICA 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} ] 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 *) nd[n_]:=Count[Range[2, n-1], _?(Abs[Mod[n, #]-#/2]<1&)]; Array[nd, 110] (* Harvey P. Dale, Jul 11 2012 *) 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)))) \\ Franklin T. Adams-Watters, 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) \\ Max Alekseyev, Apr 27 2010 (PARI) antidivisors(n)=select(t->n%t && t 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 CROSSREFS Cf. A058838, A066464, A066241, A001227, A001511. See A130799 for the anti-divisors. Sequence in context: A023135 A191654 A205784 * A237130 A058773 A122805 Adjacent sequences:  A066269 A066270 A066271 * A066273 A066274 A066275 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 31 2001 EXTENSIONS More terms from Robert G. Wilson v, Jan 02 2002 More terms from Max Alekseyev, Apr 27 2010 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 11 10:48 EST 2018. Contains 318049 sequences. (Running on oeis4.)