A013928 -id:A013928 - 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).

A373195 Cardinality of the largest subset of {1,...,n} such that no six distinct elements of this subset multiply to a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 8, 9, 9, 10, 10, 10, 10, 11, 12, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 18, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 27, 27, 28, 29, 29, 29, 29, 29, 30, 30, 30

Offset: 1

Terence Tao, May 27 2024

nonn

a(n) >= A000720(n) + A000720(n/2).
a(n) ~ 3n/2log n (Erdős-Sárközy-Sós). Best bounds currently are due to Pach-Vizer.
a(n+1)-a(n) is either 0 or 1 for any n. (Is equal to 1 when n+1 is prime.)
If "six" is replaced by "one", "two", "three", "four", "five", or "any odd", one obtains A028391, A013928, A372306, A373119, A373178, and A373114 respectively.

			a(9)=8, because {1,2,3,4,5,7,8,9} does not contain six distinct elements that multiply to a square, but {1,2,3,4,5,6,7,8,9} has 1*2*3*4*6*9 = 36^2.

P. Erdős, A. Sárközy, and V. T. Sós, On Product Representations of Powers, I, Europ. J. Combinatorics 16 (1995), 567-588.
P. Pach and M. Vizer, Improved Lower Bounds for Multiplicative Square-Free Sequences, The Electronic Journal of Combinatorics, Volume 30, Issue 4 (2023), P4.31.
Terence Tao, On product representations of squares, arXiv:2405.11610 [math.NT], May 2024.

Similar to A013928, A028391, A372306, A373114, A373119, A373178.

Cf. A000720.

from math import isqrt
def is_square(n):
    return isqrt(n) ** 2 == n
def precompute_tuples(N):
    tuples = []
    for i in range(1, N + 1):
        for j in range(i + 1, N + 1):
            for k in range(j + 1, N + 1):
                for l in range(k + 1, N + 1):
                    for m in range(l + 1, N + 1):
                        for n in range(m + 1, N + 1):
                            if is_square(i * j * k * l * m * n):
                                tuples.append((i, j, k, l, m, n))
    return tuples
def valid_subset(A, tuples):
    set_A = set(A)
    for i, j, k, l, m, n in tuples:
        if i in set_A and j in set_A and k in set_A and l in set_A and m in set_A and n in set_A:
            return False
    return True
def largest_subset_size(N, tuples):
    from itertools import combinations
    for size in reversed(range(1, N + 1)):
        for subset in combinations(range(1, N + 1), size):
            if valid_subset(subset, tuples):
                return size
for N in range(1, 26):
    print(largest_subset_size(N, precompute_tuples(N)))

from math import prod
from itertools import combinations
from functools import lru_cache
from sympy.ntheory.primetest import is_square
@lru_cache(maxsize=None)
def A373195(n):
    if n==1: return 1
    i = A373195(n-1)+1
    if sum(1 for p in combinations(range(1,n),5) if is_square(n*prod(p))) > 0:
        a = [set(p) for p in combinations(range(1,n+1),6) if is_square(prod(p))]
        for q in combinations(range(1,n),i-1):
            t = set(q)|{n}
            if not any(s<=t for s in a):
                return i
        else:
            return i-1
    else:
        return i # Chai Wah Wu, May 30 2024

From David A. Corneth, May 29 2024: (Start)
a(p) = a(p-1) + 1 for prime p.
a(k^2) = a(k^2 - 1) for k >= 3. (End)

a(26)-a(27) from Paul Muljadi, May 28 2024
a(28)-a(35) from Michael S. Branicky, May 29 2024
a(36)-a(37) from David A. Corneth, May 29 2024
a(38)-a(69) from Jinyuan Wang, Dec 30 2024

A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.

Original entry on oeis.org

1, 4, 8, 2, 9, 12, 16, 3, 5, 18, 20, 6, 24, 25, 27, 7, 28, 10, 32, 11, 36, 40, 44, 13, 14, 45, 15, 17, 48, 49, 50, 19, 52, 54, 56, 21, 60, 63, 64, 22, 68, 72, 75, 23, 26, 76, 80, 29, 30, 31, 81, 33, 84, 34, 88, 35, 90, 92, 96, 37, 98, 99, 38, 39, 100, 104, 108, 41, 112, 116

Offset: 1

Amarnath Murthy, Oct 16 2003

nonn

From Antti Karttunen, Jun 04 2014: (Start)
This is a self-inverse permutation (involution) of natural numbers.
After 1, nonsquarefree numbers occur (in monotonic order) at the positions given by squarefree numbers, A005117, and squarefree numbers occur (in monotonic order) at the positions given by their complement, nonsquarefree numbers, A013929.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Cf. A026239, A088610-A243352, A243347.

From Antti Karttunen, Jun 04 2014: (Start)
a(1), and for n>1, if mu(n) = 0, a(n) = A005117(1+A057627(n)), otherwise, a(n) =  A013929(A013928(n)). [Here mu is Moebius mu-function, A008683, which is zero only when n is a nonsquarefree number, one of the numbers in A013929].
For all n > 1, A008966(a(n)) = 1 - A008966(n), or equally, mu(a(n)) + 1 = mu(n) modulo 2. [A property shared with A243347].
(End)

More terms from Ray Chandler, Oct 18 2003

A158819 a(n) = (number of squarefree numbers <= n) minus round(n/zeta(2)).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 1

Daniel Forgues, Mar 27 2009

sign

Race between the number of squarefree numbers and round(n/zeta(2)).

First term < 0: a(172) = -1.

G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math., 48 (1917), pp. 76-92.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition, Clarendon Press, 1979, pp. 269-270.

Daniel Forgues, Table of n, a(n) for n=1..100000
Andrew Granville, ABC allows us to count squarefrees, International Mathematics Research Notices, Vol. 1998, No. 19 (1998), pp. 991-1109; alternative link.

Cf. A008966 (1 if n is squarefree, else 0).
Cf. A013928 (number of squarefree numbers < n).
Cf. A100112 (if n is the k-th squarefree number then k else 0).
Cf. A057627 (number of nonsquarefree numbers not exceeding n).
Cf. A005117 (squarefree numbers).
Cf. A013929 (nonsquarefree numbers).
Cf. A013661 (zeta(2)).

seq[lim_] := Accumulate[Boole[SquareFreeQ /@ Range[lim]]] - Round[Range[lim]/Zeta[2]]; seq[105] (* Amiram Eldar, Jan 20 2025 *)

Since zeta(2) = Sum_{i>=1} 1/(i^2) = (Pi^2)/6, we get:
a(n) = A013928(n+1) - n/Sum_{i>=1} 1/(i^2) = O(sqrt(n));
a(n) = A013928(n+1) - 6*n/(Pi^2) = O(sqrt(n)).

A238645 Number of odd primes p < 2n such that the number of squarefree numbers among 1, ..., ((p-1)/2)n is prime.

Original entry on oeis.org

0, 1, 2, 1, 2, 2, 1, 2, 2, 5, 2, 3, 3, 1, 6, 5, 3, 3, 1, 4, 2, 4, 4, 3, 4, 2, 4, 3, 1, 4, 3, 3, 7, 5, 4, 5, 5, 4, 3, 2, 5, 2, 2, 4, 5, 4, 9, 7, 4, 3, 2, 4, 3, 4, 3, 2, 4, 6, 5, 6, 4, 4, 2, 2, 7, 5, 6, 6, 8, 3, 7, 3, 5, 6, 10, 6, 6, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Mar 02 2014

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 4, 7, 14, 19, 29.
This is an analog of the conjecture in A237578 for squarefree numbers. We have verified it for n up to 20000.
See also A238646 for a similar conjecture involving squarefree numbers.

			a(4) = 1 since 3 is prime and there are exactly 3 squarefree numbers among 1, ..., (3-1)/2*4 (namely, 1, 2, 3).
a(14) = 1 since 5 and 17 are both prime, and there are exactly 17 squarefree numbers among 1, ..., (5-1)/2*14.
a(19) = 1 since 3 and 13 are both prime, and there are exactly 13 squarefree numbers among 1, ..., (3-1)/2*19 (namely, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19).
a(29) = 1 since 41 and 353 are both prime, and there are exactly 353 squarefree numbers among 1, ..., (41-1)/2*29 = 580.

Zhi-Wei Sun, Table of n, a(n) for n = 1..900
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014.

Cf. A000040, A005117, A013928, A237578, A237615, A238646.

s[n_]:=Sum[If[SquareFreeQ[k],1,0],{k,1,n}]
a[n_]:=Sum[If[PrimeQ[s[(Prime[k]-1)/2*n]],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

A275390 Numbers n for which |n/zeta(2) - Q(n)| sets a new record, where Q(x) is the number of squarefree numbers up to x.

Original entry on oeis.org

1, 2, 3, 6, 7, 15, 23, 39, 42, 43, 115, 223, 231, 239, 474, 719, 1367, 1403, 1406, 1407, 1410, 1411, 1419, 1646, 1659, 1662, 1663, 3423, 8810, 8818, 8819, 8822, 8823, 8915, 9239, 9242

Offset: 1

Charles R Greathouse IV, Aug 07 2016

nonn

Assuming the Riemann hypothesis, Vaidya proved that Q(n) = 6*n/Pi^2 + O(n^k) for any k > 2/5.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000
A. M. Vaidya, On the order of the error function of the square free numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), pp. 196-201.

Cf. A005117, A013928.

ct=r=0; for(n=1,1e4, if(issquarefree(n),ct++); t=abs(n/zeta(2)-ct); if(t>r,r=t; print1(n", ")))

A285879 Number of odd squarefree numbers <= n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 35

Offset: 1

Ilya Gutkovskiy, Apr 27 2017

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree

Cf. A005117, A008683, A013928, A056911, A185199, A285881.

ListTools:-PartialSums(map(op,[seq(`if`(numtheory:-issqrfree(n),[1,0],[0,0]),n=1..100,2)])); # Robert Israel, May 07 2018
seq(add(mobius(2*k)^2, k=1..n), n=1..100); # Ridouane Oudra, Aug 16 2019

Table[Sum[Boole[OddQ[k] && SquareFreeQ[k]], {k, 1, n}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[OddQ[k] && MoebiusMu[k]^2 == 1] x^k/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x]]

a(n) = sum(k=1, n, (k%2)*issquarefree(k)); \\ Michel Marcus, Apr 27 2017

from sympy.ntheory.factor_ import core
def a(n): return sum([1 for k in range(1, n + 1) if k%2==1 and core(k)==k]) # Indranil Ghosh, Apr 28 2017

G.f.: Sum_{k>=1} x^A056911(k)/(1 - x).
a(n) ~ 4*n/Pi^2. See A185199.
a(n) = Sum_{k=1..n} A008683(2k)^2. - Ridouane Oudra, Aug 16 2019

A295101 Number of squarefree sqrt(n)-smooth numbers <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 1

Max Alekseyev, Nov 14 2017

a(n) = number of positive squarefree integers m<=n such that A006530(m) <= sqrt(n).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A005117, A013928, A295084.

N:= 200: # for a(1)..a(N)
V:= Vector(N,1):
for n from 2 to N do
   if not numtheory:-issqrfree(n) then next fi;
   m:= max(max(numtheory:-factorset(n))^2,n);
   if m <= N then V[m..N]:= map(`+`,V[m..N],1) fi;
od:
convert(V,list); # Robert Israel, Mar 24 2020

a(n) = A013928(n+1) - Sum_{prime p > sqrt(n)} A013928(floor(n/p)+1).

If n is in A295102, then a(n)=a(n-1)+1; if n is in A001248, i.e., n=p^2 for prime p, then a(n)=a(n-1)+A013928(p); otherwise a(n)=a(n-1).

A378615 Number of non prime powers <= prime(n).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 7, 10, 13, 14, 18, 21, 22, 25, 29, 34, 35, 39, 42, 43, 48, 50, 55, 62, 65, 66, 69, 70, 73, 84, 86, 91, 92, 101, 102, 107, 112, 115, 119, 124, 125, 134, 135, 138, 139, 150, 161, 164, 165, 168, 173, 174, 182, 186, 191, 196, 197, 202, 205

Offset: 1

Gus Wiseman, Dec 06 2024

nonn

			The non prime powers counted under each term:
  n=1  n=2  n=3  n=4  n=5  n=6  n=7  n=8  n=9  n=10
  -------------------------------------------------
   1    1    1    6   10   12   15   18   22   28
                  1    6   10   14   15   21   26
                       1    6   12   14   20   24
                            1   10   12   18   22
                                 6   10   15   21
                                 1    6   14   20
                                      1   12   18
                                          10   15
                                           6   14
                                           1   12
                                               10
                                                6
                                                1

Restriction of A356068 (first-differences A143731).
First-differences are A368748.
Maxima are A378616.
Other classes of numbers (instead of non prime powers):
- prime: A000027 (diffs A000012), restriction of A000720 (diffs A010051)
- squarefree: A071403 (diffs A373198), restriction of A013928 (diffs A008966)
- nonsquarefree: A378086 (diffs A061399), restriction of A057627 (diffs A107078)
- prime power: A027883 (diffs A366833), restriction of A025528 (diffs A010055)
- composite: A065890 (diffs A046933), restriction of A065855 (diffs  A005171)
A000040 lists the primes, differences A001223
A000961 and A246655 list the prime powers, differences A057820.
A024619 lists the non prime powers, differences A375735, seconds A376599.
A080101 counts prime powers between primes (exclusive), inclusive A366833.
A361102 lists the non powers of primes, differences A375708.
Cf. A027883, A053607, A304521, A343249, A345531, A377057, A377281, A377286, A377289, A377703, A377781, A378032.

Table[Length[Select[Range[Prime[n]],Not@*PrimePowerQ]],{n,100}]

from sympy import prime, primepi, integer_nthroot
def A378615(n): return int((p:=prime(n))-n-sum(primepi(integer_nthroot(p,k)[0]) for k in range(2,p.bit_length()))) # Chai Wah Wu, Dec 07 2024

a(n) = prime(n) - A027883(n). - Chai Wah Wu, Dec 08 2024

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

cp's OEIS Frontend

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

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A373195 Cardinality of the largest subset of {1,...,n} such that no six distinct elements of this subset multiply to a square.

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A088609 a(1) = 1, a(n) is the smallest squarefree number not included earlier if n is not squarefree, else n is the smallest nonsquarefree number.

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A158819 a(n) = (number of squarefree numbers <= n) minus round(n/zeta(2)).

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A238645 Number of odd primes p < 2*n such that the number of squarefree numbers among 1, ..., ((p-1)/2)*n is prime.

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A275390 Numbers n for which |n/zeta(2) - Q(n)| sets a new record, where Q(x) is the number of squarefree numbers up to x.

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PARI

A285879 Number of odd squarefree numbers <= n.

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Maple

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A295101 Number of squarefree sqrt(n)-smooth numbers <= n.

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A238645 Number of odd primes p < 2n such that the number of squarefree numbers among 1, ..., ((p-1)/2)n is prime.