A342451 -id:A342451 - cp's OEIS frontend

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

Amiram Eldar, Mar 12 2021

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).

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

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 217.

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.

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

Cf. A005117, A004709, A046100.

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).

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

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Mar 12 2021

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.

			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.

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.

Cf. A005117, A342450, A342451.

A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

Original entry on oeis.org

2, 3, 15, 35, 231, 5005, 255255, 1616615, 10140585, 462120945, 6685349671, 1236789689135, 30425026352721, 311494317420715, 13367169186706335, 1253429172199617105, 33151040519900217915, 3909612711980232366109, 119065478046670712967865, 7970583287524270870963077

Offset: 1

Amiram Eldar, Jul 26 2022

See A356093 for details.

Amiram Eldar, Table of n, a(n) for n = 1..351

Cf. A002110, A356093 (numerators).

Similar sequences: A038111, A338560, A340819, A341432, A342451, A342480.

primorial[n_] := Product[Prime[i], {i, 1, n}]; Denominator[Table[(Prime[i] - 1)/primorial[i], {i, 1, 20}]]

a(n) = denominator((prime(n)-1)/factorback(primes(n))); \\ Michel Marcus, Jul 26 2022

from math import gcd
from sympy import primorial, prime
def A356094(n): return (p:=primorial(n))//gcd(p,prime(n)-1) # Chai Wah Wu, Jul 26 2022

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

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

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A356094 a(n) = denominator((prime(n)-1)/prime(n)#), where prime(n)# = A002110(n) is the n-th primorial.

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