A069928 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Number of arithmetic numbers <= n, cf. A003601; partial sums of A245656. - Reinhard Zumkeller, Jul 28 2014

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.

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

MATHEMATICA

Accumulate[Table[If[Divisible[DivisorSigma[1, n], DivisorSigma[0, n]], 1, 0], {n, 80}]] (* Harvey P. Dale, Oct 06 2020 *)

PROG

(PARI) for(n=1, 150, print1(sum(i=1, n, if(sigma(i)%numdiv(i), 0, 1)), ", "))

(Haskell)

a069928 n = a069928_list !! (n-1)

a069928_list = scanl1 (+) a245656_list

-- Reinhard Zumkeller, Jul 28 2014

CROSSREFS

Cf. A000005, A000203, A003601, A245656.

Sequence in context: A342880 A337774 A371263 * A135585 A081094 A361674

Adjacent sequences: A069925 A069926 A069927 * A069929 A069930 A069931

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, May 05 2002

STATUS

approved