A342452 a(n) is the least number at which the Schnirelmann density of the n-free numbers is attained.

176, 378, 2512, 3168, 31360, 236288, 1174528, 7814151, 48833536, 293001216, 1709645824, 12207734784, 67143319552

Offset: 2

Author

Amiram Eldar, Mar 12 2021

Keywords

• nonn
• more

Comments

k-free numbers are numbers whose exponents in their prime factorization are all less than k. 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. See A342450 for more information.

The value of m(k) in which Q_k(m)/m = d(k) is not necessarily unique: while for k = 2, 3 and 4 the density is attained at a single value, i.e., 176, 378 and 2512, respectively, for k = 5 the density is attained at both 3168 and 6336. Hardy (1979) found that also for k = 38, 55 and 56 the value of m(k) is not unique, and for k = 38 the density is attained in at least 3 values.

Orr (1969) proved that 5^n <= a(n) < 6^n, for n >= 5.

Diananda and Subbarao (1977) proved that the largest value of m at which the density is attained is in the interval [6^n/2, 6^n).

Hardy (1969) calculated the least value of m in this interval, for n = 2..75, but his values are not necessarily the least nor the largest.

The terms in the data section for n=2..14 were verified to be the least values. Except for n=5, they are also unique values.

Examples

			The number of squarefree numbers (A005117) up to 176 is Q_2(176) = 106. It is where the Schnirelmann density inf_{m>=1} Q_2(m)/m = 106/176 = 53/88 is attained. Therefore a(2) = 176.
		

Links

• 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.
• 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.

Crossrefs

Cf. A005117, A342450, A342451.