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 A066417 Sum of anti-divisors of n. 67
 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 (list; graph; refs; listen; history; text; internal format)
 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, Anti-divisors [Broken link] 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 EXAMPLE For example, n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {3,7,35}, {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. A066416, A066418, A058838, A064277. Sequence in context: A075158 A215526 A246841 * A254669 A227913 A079521 Adjacent sequences:  A066414 A066415 A066416 * A066418 A066419 A066420 KEYWORD nonn AUTHOR Jon Perry, Dec 28 2001 STATUS approved

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Last modified February 27 06:36 EST 2020. Contains 332299 sequences. (Running on oeis4.)