A066417 - OEIS

%I #76 Jun 14 2024 08:49:06

%S 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,

%T 24,42,46,24,36,42,58,48,30,52,32,50,70,52,55,41,66,56,40,86,58,60,56,

%U 72,80,42,94,88,52,74,56,74,96,90,107,57,78,112,46,84,86,132,112,54,102

%N Sum of anti-divisors of n.

%C See A066272 for definition of anti-divisor.

%H Michael De Vlieger, <a href="/A066417/b066417.txt">Table of n, a(n) for n = 1..10000</a>

%H Jon Perry, <a href="/A066272/a066272a.html">The Anti-divisor</a>. [Cached copy]

%H Jon Perry, <a href="/A066272/a066272.html">The Anti-divisor: Even More Anti-Divisors</a>. [Cached copy]

%F 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

%F 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

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

%e 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.

%p # Uses antidivisors() implemented in A066272.

%p A066417 := proc(n) add(d,d=antidivisors(n)) ; end proc: # _R. J. Mathar_, Jul 04 2011

%p # faster alternative with Alekseyev formula

%p A066417 := proc(n)

%p     k := A007814(n) ;

%p     numtheory[sigma](2*n-1)+numtheory[sigma](2*n+1) +numtheory[sigma(n/2^k)*2^(k+1) -6*n-2 ;

%p end proc: # _R. J. Mathar_, Nov 11 2014

%t 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 *)

%t 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 *)

%o (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

%o (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

%o (Python)

%o from sympy import divisors

%o 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

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

%K nonn

%O 1,3

%A _Jon Perry_, Dec 28 2001