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

1,3

Comments

See A066272 for definition of anti-divisor.

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]

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A075158 A215526 A246841 * A347348 A254669 A227913

Adjacent sequences: A066414 A066415 A066416 * A066418 A066419 A066420

Keyword

nonn

Author

Jon Perry, Dec 28 2001

Status

approved