A069928 Number of integers k, 1<=k<=n, such that tau(k) divides sigma(k) where tau(k) is the number of divisors of k and sigma(k) the sum of divisors of k.
1, 1, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 12, 13, 14, 15, 15, 15, 15, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 29, 30, 31, 31, 32, 32, 33, 33, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 42, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50, 50
Offset: 1
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- Paul T. Bateman, Paul Erdős, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer, in: Marvin I. Knopp (ed.), Analytic number theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol. 899, Springer-Verlag, 1981, pp. 197-220; alternative link.
- Amiram Eldar, Plot of a(n)/n for n = 2^(10..33).
- Wikipedia, Arithmetic number.
Programs
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Haskell
a069928 n = a069928_list !! (n-1) a069928_list = scanl1 (+) a245656_list -- Reinhard Zumkeller, Jul 28 2014
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Mathematica
Accumulate[Table[If[Divisible[DivisorSigma[1,n],DivisorSigma[0,n]],1,0],{n,80}]] (* Harvey P. Dale, Oct 06 2020 *) -
PARI
for(n=1,150,print1(sum(i=1,n,if(sigma(i)%numdiv(i),0,1)),","))
Formula
a(n) = Card(k: 1<=k<=n : sigma(k) == 0 (mod tau(k))).
Limit_{n -> infinity} a(n)/n = C = 0.8...
Bateman et al. (1981) proved that the asymptotic density of the arithmetic numbers is 1. Therefore, the formula above is correct, but limit is C = 1. - Amiram Eldar, Dec 28 2024
Comments