OFFSET
2,1
COMMENTS
k-free numbers are numbers whose exponents in their prime factorization are all less than k. E.g., the squarefree numbers (k=2, A005117), the cubefree numbers (k=3, A004709) and the biquadratefree numbers (k=4, A046100).
Let Q_k(m) be the number of k-free numbers not exceeding m. The Schnirelmann density for k-free numbers is d(k) = inf_{m>=1} Q_k(m)/m.
a(2) was found by Rogers (1964).
a(3)-a(6) were found by Orr (1969).
a(7)-a(75) were found by Hardy (1979).
REFERENCES
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217.
LINKS
Amiram Eldar, Table of n, a(n) for n = 2..75 (from Hardy, 1979)
P. H. Diananda and M. V. Subbarao, On the Schnirelmann density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (1977), pp. 7-10.
R. L. Duncan, The Schnirelmann density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 16, No. 5 (1965), pp. 1090-1091.
R. L. Duncan, On the density of the k-free integers, Fibonacci Quarterly, Vol. 7, No. 2 (1969), pp. 140-142.
Paul Erdős, G. E. Hardy and M. V. Subbarao, On the Schnirelmann density of k-free integers, Indian J. Math., Vol. 20 (1978), pp. 45-56.
George Eugene Hardy, On the Schnirelmann density of the k-free and (k,r)-free integers, Ph.D. thesis, University of Alberta, 1979.
Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
Kenneth Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the American Mathematical Society, Vol. 15, No. 4 (1964), pp. 515-516.
Harold M. Stark, On the asymptotic density of the k-free integers, Proceedings of the American Mathematical Society, Vol. 17, No. 5 (1966), pp. 1211-1214.
M. V. Subbarao, On the Schnirelman density of the K-free integers, Distribution of values of arithmetic functions, Vol. 517 (1984), pp. 47-61; alternative link.
Eric Weisstein's World of Mathematics, Schnirelmann Density.
Wikipedia, Schnirelmann density.
FORMULA
Let d(n) = a(n)/A342451(n), and let D(n) = 1/zeta(n), the asymptotic density of the n-free numbers. Then:
Lim_{n->oo} d(n) = 1.
d(n) < D(n) (Stark, 1966).
d(n) < D(n) < d(n+1) < D(n+1) (Duncan, 1965; Erdős et al., 1978).
d(n) > 1 - Sum_{p prime} 1/p^n (Duncan, 1969).
(D(n+1)-d(n+1))/(D(n)-d(n)) < 1/2^n (Duncan, 1969).
d(n) > 1 - 1/2^n - 1/3^n - 1/5^n (Diananda and Subbarao, 1977).
EXAMPLE
The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ...
KEYWORD
nonn,frac
AUTHOR
Amiram Eldar, Mar 12 2021
STATUS
approved