A066417 Sum of anti-divisors of n.

0, 0, 2, 3, 5, 4, 10, 8, 8, 14, 12, 13, 19, 16, 18, 14, 28, 28, 18, 24, 22, 36, 34, 23, 39, 24, 42, 46, 24, 36, 42, 58, 48, 30, 52, 32, 50, 70, 52, 55, 41, 66, 56, 40, 86, 58, 60, 56, 72, 80, 42, 94, 88, 52, 74, 56, 74, 96, 90, 107, 57, 78, 112, 46, 84, 86, 132, 112, 54, 102

Offset: 1

Author

Jon Perry, Dec 28 2001

Keywords

• nonn

Comments

See A066272 for definition of anti-divisor.

Examples

			For 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 12 are 4, 5, 7, 12. Therefore a(18) = 28.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Jon Perry, The Anti-divisor. [Cached copy]
• Jon Perry, The Anti-divisor: Even More Anti-Divisors. [Cached copy]

Crossrefs

Cf. A000203, A007814, A066272, A066416, A066418, A058838, A064277.

Programs

Maple

# Uses antidivisors() implemented in A066272.
A066417 := proc(n) add(d,d=antidivisors(n)) ; end proc: # R. J. Mathar, Jul 04 2011
# faster alternative with Alekseyev formula
A066417 := proc(n)
    k := A007814(n) ;
    numtheory[sigma](2*n-1)+numtheory[sigma](2*n+1) +numtheory[sigma(n/2^k)*2^(k+1) -6*n-2 ;
end proc: # R. J. Mathar, Nov 11 2014

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[ Plus @@ antid[n], {n, 70}] (* Robert G. Wilson v, Mar 15 2004 *)
a066417[n_Integer] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]]; Array[a066417, 120] (* _Michael De Vlieger, Aug 08 2014, after Harvey P. Dale at A066272 *)

PARI

al(n)=Vec(sum(k=1,n,2*k*(x^(3*k)+x*O(x^n))/(1-x^(2*k))+(2*k+1)*(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) = my(k); if(n>1, k=valuation(n,2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0); } \\ Max Alekseyev, Apr 27 2010

Python

from sympy import divisors
A066417 = [sum([2*d for d in divisors(n) if n > 2*d and n%(2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n%d] + [d for d in divisors(2*n+1) if n > d >=2 and n%d]) for n in range(1,10**6)] # Chai Wah Wu, Aug 12 2014

Formula

G.f.: Sum_{k>0} (2k x^(3k) / (1 - x^(2k)) + (2k+1)(x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1))). - Franklin T. Adams-Watters, Sep 11 2009

For n>1, a(n) = A000203(2*n-1) + A000203(2*n+1) + A000203(n/2^k)*2^(k+1) - 6*n - 2, where k=A007814(n). - Max Alekseyev, Apr 27 2010

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/8 - 3 = 0.70110165... . - Amiram Eldar, Jan 19 2024