A078709 - OEIS

%I #21 Jun 19 2022 09:55:19

%S 1,1,1,1,2,1,3,2,3,2,5,2,6,3,3,3,8,3,9,3,5,5,11,3,8,6,6,4,14,3,15,5,8,

%T 8,8,4,18,9,9,5,20,5,21,7,7,11,23,4,16,8,12,8,26,6,13,7,14,14,29,5,30,

%U 15,10,9,16,8,33,11,17,8,35,6,36,18,12,12,19,9,39,8,16,20,41,7,21,21,21,11

%N a(n) = floor(n/d(n)), where d(n) is the number of divisors of n (A000005).

%C Also, integer part of the mean subinterval length in the partition of [0,n] by the divisors of n.

%C If the first occurrence of m in the sequence is greater than all preceding terms, the corresponding n is noncomposite. - Donald Sampson (Marsquo(AT)hotmail.com), Dec 10 2003

%H Harvey P. Dale, <a href="/A078709/b078709.txt">Table of n, a(n) for n = 1..1000</a>

%e The divisors of 9 partition the closed interval [0,9] into subintervals [0,1), [1,3), [3,9], with lengths 1, 2, 6, respectively. The mean of these lengths has integer part = 3. Hence a(9) = 3.

%t << Statistics`DescriptiveStatistics` f[n_] := Module[{d, l, a, i}, d = Divisors[n]; l = Length[d]; a = {1}; For[i = 1, i <= l - 1, i++, a = Append[a, d[[i + 1]] - d[[i]]]]; a]; Table[Floor[Mean[f[i]]], {i, 1, 100}]

%t Table[Floor[n/DivisorSigma[0,n]],{n,90}] (* _Harvey P. Dale_, Jun 10 2016 *)

%o (Python)

%o from sympy import divisor_count

%o def A078709(n): return n//divisor_count(n) # _Chai Wah Wu_, Jun 03 2022

%Y Cf. A000005, A078710.

%K nonn

%O 1,5

%A _Joseph L. Pe_, Dec 19 2002

%E Replaced definition with a simpler definition suggested by _Reinhard Zumkeller_, Feb 26 2003. The original definition is now a comment. - _N. J. A. Sloane_, Jun 19 2022