A123321 -id:A123321 - cp's OEIS frontend

A077438 Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

4, 9, 25, 49, 121, 169, 289, 361, 529, 841, 900, 961, 1369, 1681, 1764, 1849, 2209, 2809, 3481, 3721, 4356, 4489, 4900, 5041, 5329, 6084, 6241, 6889, 7921, 9409, 10201, 10404, 10609, 11025, 11449, 11881, 12100, 12769, 12996, 16129, 16900

Offset: 1

Benoit Cloitre, Nov 30 2002

nonn

From Robert G. Wilson v, Dec 28 2016: (Start)
Union of {A000040, A007304, A046387, A123321, A115343, etc}^2 = Union of {A001248, A162143, etc} = A030059(n)^2.
Number of terms < 10^k: 2, 4, 12, 30, 98, 303, 957, ..., . (End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1000 (first 800 terms from G. C. Greubel)

Cf. A000040, A001248, A007304, A030059, A046387, A088245, A115343, A123321, A162143.

fQ[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[#] MoebiusMu[n/#]^2 & /@ d) == -1]; Select[Range@17000, fQ] (* Robert G. Wilson v, Dec 28 2016 *)

isok(n) = sumdiv(n, d, moebius(d)*moebius(n/d)^2) == -1; \\ Michel Marcus, Nov 08 2013

is(n)=if(!issquare(n,&n), return(0)); my(f=factor(n)[,2]); #f%2 && vecmax(f)==1 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A030059(n)^2.
From Amiram Eldar, Jun 16 2020: (Start)
Sum_{k>=1} 1/a(k) = 9/(2*Pi^2) = A088245.
Sum_{k>=1} 1/a(k)^2 = 15/(2*Pi^4). (End)

A327829 Squarefree numbers with a prime number of prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Sebastian F. Orellana, Sep 26 2019

nonn

210 is the first integer in A120944 but not here: it has 4 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A120944.
A006881, A007304, A046387 are subsequences.
A046386, A067885 are not subsequences.

Select[Range@ 161, And[SquareFreeQ@ #, PrimeQ@ PrimeNu@ #] &] (* Michael De Vlieger, Sep 29 2019 *)

isok(n) = issquarefree(n) && isprime(omega(n)); \\ Michel Marcus, Sep 27 2019

A006881 UNION A007304 UNION A046387 UNION A123321 UNION .... - R. J. Mathar, Oct 13 2019

Corrected and extended by Michel Marcus, Sep 27 2019

A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

Original entry on oeis.org

2, 39, 1279786, 8377774397163159586

Offset: 1

Jon E. Schoenfield, Nov 22 2022

			The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786. There are 75 numbers with this property, the last one being 1281378.
At k=8377774397163159586, the interval [1..k] includes 1356557942402075858 squarefree numbers with 3 prime factors and the same number of squarefree numbers with 4 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(4)=8377774397163159586. There are 14 numbers with this property, the last one being 8377774397163162544. - _Henri Lifchitz_, Jan 31 2025

Cf. A005117, A072047.
Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.
Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.
Cf. A340316.

a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A356683(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(k,n): return sum(primepi(k//prod(c[1] for c in a))-a[-1][0] for a in g(k,0,1,1,n)) if n>1 else primepi(k)
    return 2 if n==1 else next(k for k in count(1) if (x:=f(k,n-1))>0 and x==f(k,n)) # Chai Wah Wu, Aug 31 2024

a(4) from Henri Lifchitz, Jan 31 2025

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A077438 Numbers k such that Sum_{d|k} mu(d) mu(n/d)^2 = -1.

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A327829 Squarefree numbers with a prime number of prime factors.

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A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

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