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
Keywords
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]
Programs
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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
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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[ 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
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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
Comments