A342450 a(n) is the numerator of the Schnirelmann density of the n-free numbers.

53, 157, 145, 3055, 6165, 234331, 584879, 2599496, 48785015, 292856489, 854612603, 12206236915, 8392400925, 183100803621, 1296977891119, 15258697717317, 2997253335821, 79472769236347, 556309528064071, 5960463317677243, 25033951904190895, 46938653648975843, 3099441423652148001

Offset: 2

Author

Amiram Eldar, Mar 12 2021

Keywords

• nonn
• frac

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).

Examples

			The fractions begin with 53/88, 157/189, 145/157, 3055/3168, 6165/6272, 234331/236288, 584879/587264, 2599496/2604717, 48785015/48833536, 292856489/293001216, ...
		

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.

Crossrefs

Cf. A013928, A336025, A342451 (denominators), A342452.

Cf. A005117, A004709, A046100.

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).