A102722 - OEIS

%I #21 Oct 25 2023 09:30:35

%S 0,0,0,0,1,0,2,1,2,2,4,2,4,4,4,4,6,4,7,5,6,7,9,6,8,9,10,8,11,8,11,10,

%T 11,13,14,10,13,14,15,13,16,13,17,16,15,17,20,16,18,17,19,18,22,20,21,

%U 19,20,22,26,19,23,25,24,23,25,23,26,26,28,26,30,23,27,29,29,29,31,29,33

%N Given n, sum all division remainders {n/k}, with k=1,...,n. The value a(n) is given by the floor of that sum. Note that {x}:=x-[x].

%C Conjecture: a(n) ~ (1-EulerGamma)n.

%H Robert Israel, <a href="/A102722/b102722.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = floor(n*H(n)) - Sum_{j=1..n} d(j), where d(n)=A000005(n) is the number of divisors of n, and H(n) is the n-th Harmonic Number. [_Enrique Pérez Herrero_, Aug 25 2009; corrected by _Robert Israel_, Mar 20 2016]

%F a(n) = A052488(n) - A006218(n). [_Enrique Pérez Herrero_, Aug 25 2009]

%e a(5) = [{5/1}+{5/2}+{5/3}+{5/4}+{5/5}]=[0+0.5+0.6666+0.25+0]=[1.4166]=1 (division by 1 or by the number itself is to be avoided).

%p N:= 100:

%p H:= ListTools:-PartialSums([seq(1/n,n=1..N)]):

%p S:= ListTools:-PartialSums(map(numtheory:-tau,[$1..N])):

%p seq(floor(n*H[n])-S[n],n=1..N); # _Robert Israel_, Mar 20 2016

%t Resto = Function[n, Sum[n/k - Floor[n/k], {k, 2, n - 1}]]; Floor[Map[Resto, Range[1, 1000]]]

%t Table[Floor[n*HarmonicNumber[n]] - Sum[DivisorSigma[0, k], {k, 1, n}], {n, 1, 200}] (* _Enrique Pérez Herrero_, Aug 25 2009 *)

%t Table[Floor[Sum[FractionalPart[n/k], {k, 1, n}]], {n, 1, 200}] (* _Enrique Pérez Herrero_, Aug 25 2009 *)

%o (Python)

%o from math import isqrt

%o from sympy import harmonic

%o def A102722(n): return int(n*harmonic(n))+(s:=isqrt(n))**2-(sum(n//k for k in range(1,s+1))<<1) # _Chai Wah Wu_, Oct 24 2023

%Y Cf. A052488, A006218.

%Y Cf. A000005.

%K easy,nonn

%O 1,7

%A _Carlos Alves_, Feb 06 2005