A034444 -id:A034444 - cp's OEIS frontend

A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).

8, 24, 27, 40, 54, 56, 88, 104, 120, 125, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 343, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696

Offset: 1

Labos Elemer

nonn

For these terms the number of divisors should be a special power of two because ud(n) = 2^r and nud(n) = ud(n). In particular the exponent of 2 is 1+A001221(n), the number of distinct prime factors + 1. Thus this is a subsequence of A036537 where A000005(A036537(n)) = 2^s; here s = 1+A001221(n).
Let us introduce a function D(n) = sigma_0(n)/2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n = p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A001222). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (A005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A060687. - Ctibor O. Zizka, Sep 21 2008
Integers n such that there are exactly 3 Abelian groups of order n. That is, n such that A000688(n)=3. In other words, in the prime factorization of n there is exactly one prime with exponent of 3 and the others have exponent of 1. - Geoffrey Critzer, Jun 09 2015
The asymptotic density of this sequence is (6/Pi^2) * Sum_{k>=1} 1/(prime(k)^2*(prime(k)+1)) = (1/zeta(2)) * Sum_{k>=3} (-1)^(k+1) * P(k) = 0.0741777413672596019212880156082745910562809066233004356300970463709875..., where P is the prime zeta function. - Amiram Eldar, Jul 11 2020

			n = 88 = 2*2*2*11 has 8 divisors, of which 4 are unitary divisors (because of 2 distinct prime factors) and 4 are nonunitary divisors: U={1,88,11,8} and NU = {2,44,4,22}.

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

Cf. A000005, A001221, A034444, A036537, A048106, A048107.

filter:= proc(n) local F;
  F:= ifactors(n)[2];
  mul(t[2]+1,t=F) = 2^(1+nops(F))
end proc;
select(filter, [$1..1000]); # Robert Israel, Jun 09 2015

Position[Table[FiniteAbelianGroupCount[n], {n, 1, 1000}],3] // Flatten (* Geoffrey Critzer, Jun 09 2015 *)

is(n)=select(e->e>1, factor(n)[,2])==[3]~ \\ Charles R Greathouse IV, Jun 10 2015

isok(n) = sumdiv(n, d, issquarefree(d)) == sumdiv(n, d, !issquarefree(d)); \\ Michel Marcus, Jun 24 2015

from math import isqrt
from sympy import mobius, primerange
def A048109(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x+sum(sum(-g(x//p**j) if j&1 else g(x//p**j) for j in range(3,x.bit_length())) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

Numbers k such that d(k) = 2^(omega(k)+1) or A000005(k) = 2^(A001221(k) + 1) = 2 * A034444(k).

New name based on comment by Ivan Neretin, Jun 19 2015

A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5

Offset: 1

Geoffrey Critzer, Jul 09 2015

nonn

Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups.  A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.
Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019

			a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2. - _Antti Karttunen_, Oct 21 2017

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Hall subgroup
Index entries for sequences computed from exponents in factorization of n

Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.
Cf. A304716, A302569, A304711, A305079.
Related classes of factorizations:
- No conditions: A001055
- Strict: A045778
- Constant: A089723
- Distinct multiplicities: A255231
- Singleton or coprime: A259936
- Relatively prime: A281116
- Aperiodic: A303386
- Stable (indivisible): A305149
- Connected: A305193
- Strict relatively prime: A318721
- Uniform: A319269
- Intersecting: A319786
- Constant or distinct factors coprime: A327399
- Constant or relatively prime: A327400
- Coprime: A327517
- Not relatively prime: A327658
- Distinct factors coprime: A327695

map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015

Table[BellB[PrimeNu[n]], {n, 1, 75}]
(* second program *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==1||CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Sep 24 2019 *)

a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, -z*x^2, 1 - (z+1)*x)))); polcoeff(1/(1 - x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017

a(n) = A000110(A001221(n)).

a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019

Incorrect comment removed by Antti Karttunen, Jun 11 2022

A064608 Partial sums of A034444: sum of number of unitary divisors from 1 to n.

Original entry on oeis.org

1, 3, 5, 7, 9, 13, 15, 17, 19, 23, 25, 29, 31, 35, 39, 41, 43, 47, 49, 53, 57, 61, 63, 67, 69, 73, 75, 79, 81, 89, 91, 93, 97, 101, 105, 109, 111, 115, 119, 123, 125, 133, 135, 139, 143, 147, 149, 153, 155, 159, 163, 167, 169, 173, 177, 181, 185, 189, 191, 199, 201

Offset: 1

Labos Elemer, Sep 24 2001

a(n) = Sum_{k<=n} 2^omega(k) where omega(k) is the number of distinct primes in the factorization of k. - Benoit Cloitre, Apr 16 2002

a(n) is the number of (p, q) lattice points that are visible from (0, 0), where p and q satisfy: p >= 1, q >= 1, p * q <= n. - Luc Rousseau, Jul 09 2017

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909 (Chelsea reprint 1953), p. 594.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Masum Billal, Number of Ways To Write as Product of Co-prime Numbers, arXiv:1909.07823 [math.GM], 2019.
E. Cohen, The number of unitary divisors of an integer, Am. Math. Mon. 67, 879-880 (1960).
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
F. Mertens, Uber einige asymptotische Gesetze der Zahlentheorie, J. Reine Angew. Math., 77 (1874), 289-338.
V. Sitaramaiah and M.V. Subbarao, Unitary divisor problem for arithmetic progressions, Annales Univ. Sci. Budapest., Sect. Comp. 32 (2010) 73-89.
D. Suryanarayana and V. Siva Rama Prasad, The number of k-free divisors of an integer, Acta Arithmetica XVII (1971), 345-354.
D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7. Eq (8) for asymptotics.

Cf. A001620, A001221, A034444, A064610, A069212.

with(numtheory): A064608:=n->add(mobius(k)^2*floor(n/k), k=1..n): seq(A064608(n), n=1..100); # Wesley Ivan Hurt, Dec 05 2015

a[n_] := Count[Divisors@ n, d_ /; GCD[d, n/d] == 1]; Accumulate@ Array[a, {61}] (* Michael De Vlieger, Oct 21 2015, after Jean-François Alcover at A034444 *)
Accumulate@ Array[2^PrimeNu[#] &, {61}] (* Amiram Eldar, Oct 21 2019 *)

{ for (n=1, 80, a=sum(k=1, n, moebius(k)^2*floor(n/k)); write("b064608.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 20 2009

a(n)=sum(k=1,sqrtint(n),moebius(k)*(2*sum(l=1,sqrtint(n\(k*k)),n\(k*k*l))-sqrtint(n\(k*k))^2)); \\ More efficient formula for large n values (up to 10^14)
vector(80,i,a(i)) \\ Jerome Raulin, Nov 01 2015

from sympy.ntheory.factor_ import primenu
def A064608(n): return sum(1<Chai Wah Wu, Sep 07 2023

a(n) = a(n-1) + A034444(n) = a(n-1) + 2^A001221(n) Sum_{j=1..n} ud(j) where ud(j) = A034444(j) = 2^A001221(n).
a(n) = n*log(n)/zeta(2) + O(n) where zeta(2) = Pi^2/6. - Benoit Cloitre, Apr 16 2002
a(n) = Sum_{k=1..n} mu(k)^2*floor(n/k). - Benoit Cloitre, Apr 16 2002
Mertens's theorem (1874): a(n) = Sum_{k<=n} ud(k) = (n/Zeta(2))*(log(n) + 2*gamma - 1 - 2*Zeta'(2)/Zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
G.f.: (1/(1 - x))*Sum_{k>=1} mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017

A322327 a(n) = A005361(n) * A034444(n).

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 4, 4, 2, 8, 2, 4, 4, 8, 2, 8, 2, 8, 4, 4, 2, 12, 4, 4, 6, 8, 2, 8, 2, 10, 4, 4, 4, 16, 2, 4, 4, 12, 2, 8, 2, 8, 8, 4, 2, 16, 4, 8, 4, 8, 2, 12, 4, 12, 4, 4, 2, 16, 2, 4, 8, 12, 4, 8, 2, 8, 4, 8, 2, 24, 2, 4, 8, 8, 4, 8, 2, 16, 8, 4, 2, 16, 4, 4, 4, 12, 2, 16, 4, 8, 4, 4, 4, 20, 2, 8, 8, 16

Offset: 1

Werner Schulte, Dec 03 2018

Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.

Dirichlet inverse b(n) [= A355837(n)] is multiplicative with b(p^e) = 2 * (e mod 2) * (-1)^((e+1)/2) for prime p and e > 0.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A000005, A000007, A001221, A001620, A005361, A034444, A226602, A227291, A286324, A322328, A355837 (Dirichlet inverse).

a[n_] := If[n==1, 1, Module[{f = FactorInteger[n]}, 2^Length[f] * Times@@f[[;;,2]]]]; Array[a, 100] (* Amiram Eldar, Dec 03 2018 *)

a(n) = my(f=factor(n)); vecprod(f[,2])*2^omega(n); \\ Michel Marcus, Dec 04 2018

A322327(n) = factorback(apply(e -> e+e, factor(n)[, 2])); \\ Antti Karttunen, Jul 18 2022

from math import prod
from sympy import factorint
def A322327(n): return prod(e<<1 for e in factorint(n).values()) # Chai Wah Wu, Dec 26 2022

Multiplicative with a(p^e) = 2*e for prime p and e > 0.
Dirichlet g. f.: (zeta(s))^2 * zeta(2*s) / zeta(4*s).
Equals Dirichlet convolution of A000005 and A227291.
Sum_{k=1..n} a(k) ~ 15*(log(n) + 2*gamma - 1 + 12*zeta'(2)/Pi^2 - 360*zeta'(4)/Pi^4) * n / Pi^2 + 6*zeta(1/2)^2 * sqrt(n) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 20 2020
a(n) = A005361(n^2) = A286324(n^2). - Amiram Eldar, Dec 09 2023

Data section extended up to a(100) by Antti Karttunen, Jul 18 2022

A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86

Offset: 1

Labos Elemer

nonn

Numbers with at most one 2 and no 3s or higher in their prime exponents. - Charles R Greathouse IV, Aug 25 2016

A disjoint union of A005117 and A060687. The asymptotic density of this sequence is (6/Pi^2) * (1 + Sum_{p prime} 1/(p*(p+1))) = A059956 * (1 + A179119) = A059956 + A271971 = 0.8086828238... - Amiram Eldar, Nov 07 2020

			n = 420 = 2*2*3*5*7, 4 distinct prime factors, 24 divisors of which 16 are unitary and 8 are not; ud(n) > nud(n) and 2^(4+1) = 32 is larger than d, the number of divisors.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Complement of A048108.
A072357 is a subsequence.
Cf. A000005, A001221, A005117, A034444, A048105, A059956, A060687, A179119, A271971, A325973.

Select[Range[500], 2^(1 + PrimeNu[#]) > DivisorSigma[0, #] &] (* G. C. Greubel, May 05 2017 *)

is(n)=my(f=factor(n)[, 2], t); for(i=1, #f, if(f[i]>1, if(t||f[i]>2, return(0), t=1))); 1 \\ Charles R Greathouse IV, Sep 17 2015

is(n)=n==1 || factorback(factor(n)[,2])<3 \\ Charles R Greathouse IV, Aug 25 2016

Numbers for which 2^(r(n)+1) > d(n), where r = A001221, d = A000005.

A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

Original entry on oeis.org

16, 32, 36, 48, 64, 72, 80, 81, 96, 100, 108, 112, 128, 144, 160, 162, 176, 180, 192, 196, 200, 208, 216, 224, 225, 240, 243, 252, 256, 272, 288, 300, 304, 320, 324, 336, 352, 360, 368, 384, 392, 396, 400, 405, 416, 432, 441, 448, 450, 464, 468, 480, 484

Offset: 1

Labos Elemer

nonn

Numbers n that are expressible as a product of 2 "nonsquarefree" numbers (i.e., there are 2 integers x,y in A001694 such that n = xy). - Benoit Cloitre, Jan 01 2003
Also numbers having more than one square divisor > 1: A046951(a(n)) > 2. - Reinhard Zumkeller, Apr 08 2003
The asymptotic density of this sequence is 1 - (6/Pi^2)*(1 + Sum_{n>=1} 1/prime(n)^2) = 1 - A059956 * (1 + A085548) = 0.1171394347594477824... . - Amiram Eldar, Sep 25 2022

			36 is in the sequence since the number of its unitary divisors, {1, 4, 9, 36} is 4 which is smaller than 5, the number of its non-unitary divisors, {2, 3, 6, 12, 18}.

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

Cf. A000005, A001221, A034444, A048105, A048109, A082293, A013929, A082294, A082295.

Cf. A085548, A059956.

Select[Range[484], DivisorSigma[0, #] > 2^(PrimeNu[#]+1) &] (* Amiram Eldar, Jun 11 2019 *)

is(n)=my(f=factor(n)[,2],t); for(i=1,#f,if(f[i]>1, if(t||f[i]>3, return(1), t=1))); 0 \\ Charles R Greathouse IV, Sep 17 2015

A000005(a(n)) > 2^(1 + A001221(a(n))).

A048166 Numbers k that are divisible by the number of unitary divisors of k (A034444).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 196, 200, 208, 212, 216, 224, 232, 236, 240, 244, 248, 256, 264, 268

Offset: 1

Labos Elemer

nonn

			a(81) = 392 = 2^3*7^2 has 4 unitary divisors, {1, 392, 8, 49}, and 4 divides 392.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms n = 1..5000 from G. C. Greubel)

Cf. A033950, A034444, A000005, A048167, A048481.

okQ[n_] := Divisible[n, DivisorSum[n, Boole[CoprimeQ[#, n/#]]&]]; Select[ Range[300], okQ] (* Jean-François Alcover, Dec 05 2015 *)
Select[Range[270], Divisible[#, 2^PrimeNu[#]] &] (* Amiram Eldar, Jul 16 2019 *)

isok(n) = !(n % sumdiv(n, d, gcd(d, n/d)==1)); \\ Michel Marcus, Feb 25 2014

isok(n) = !(n % 2^omega(n)); \\ Amiram Eldar, Jul 16 2019

Binomial transform of [1, 1, 1, 1, -3, 5, -7, 9, -11, 13, ...]. Binomial transform of this sequence is A048481. - Gary W. Adamson, Oct 23 2007

A048106 Number of unitary divisors of n (A034444) - number of non-unitary divisors of n (A048105).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 0, 1, 4, 2, 2, 2, 4, 4, -1, 2, 2, 2, 2, 4, 4, 2, 0, 1, 4, 0, 2, 2, 8, 2, -2, 4, 4, 4, -1, 2, 4, 4, 0, 2, 8, 2, 2, 2, 4, 2, -2, 1, 2, 4, 2, 2, 0, 4, 0, 4, 4, 2, 4, 2, 4, 2, -3, 4, 8, 2, 2, 4, 8, 2, -4, 2, 4, 2, 2, 4, 8, 2, -2, -1, 4, 2, 4, 4, 4, 4, 0, 2, 4, 4, 2, 4, 4, 4, -4, 2, 2, 2

Offset: 1

Labos Elemer

sign

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A008683, A034444, A048105.
Cf. A048107, A048108, A048109, A048111.
Cf. A001620, A306016.

Table[2^(1 + PrimeNu@ n) - DivisorSigma[0, n], {n, 99}] (* Michael De Vlieger, Aug 01 2017 *)

A048106(n) = (2^(1+omega(n)) - numdiv(n)); \\ Antti Karttunen, May 25 2017

from sympy import divisor_count, primefactors
def a(n): return 1 if n==1 else 2**(1 + len(primefactors(n))) - divisor_count(n) # Indranil Ghosh, May 25 2017

a(n) = 2^(1+omega(n)) - d(n) = 2^(1+A001221(n)) - A000005(n).
a(n) = -Sum_{ d divides n } (-1)^mu(d). - Vladeta Jovovic, Jan 24 2002
From Amiram Eldar, Dec 09 2022: (Start)
a(n) > 0 iff n is in A048107.
a(n) < 0 iff n is in A048111.
a(n) <= 0 iff n is in A048108.
a(n) = 0 iff n is in A048109.
Dirichlet g.f: zeta(s)^2*(2/zeta(2*s) - 1).
Sum_{k=1..n} a(k) ~ (12/Pi^2 - 1)*n*log(n) + ((12/Pi^2-1)*(2*gamma-1) - (24/Pi^2)*zeta'(2)/zeta(2))*n, where gamma is Euler's constant (A001620). (End)

A048108 Numbers with at least as many non-unitary divisors (A048105) as unitary divisors (A034444).

Original entry on oeis.org

8, 16, 24, 27, 32, 36, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 100, 104, 108, 112, 120, 125, 128, 135, 136, 144, 152, 160, 162, 168, 176, 180, 184, 189, 192, 196, 200, 208, 216, 224, 225, 232, 240, 243, 248, 250, 252, 256, 264, 270, 272, 280, 288, 296, 297

Offset: 1

N. J. A. Sloane, Labos Elemer

nonn

Numbers divisible by a prime cubed or two distinct primes squared. - Charles R Greathouse IV, Jun 07 2013

Equals A013929 \ A060687. The asymptotic density of this sequence is 1 - A059956 - A271971 = 0.1913171761... - Amiram Eldar, Nov 07 2020

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Complement of A048107.
Subsequence of A013929.
Cf. A059956, A060687, A271971.

with(numtheory): for n from 1 to 800 do if 2^nops(ifactors(n)[2])<=tau(n)-2^nops(ifactors(n)[2]) then printf(`%d,`,n) fi; od:

Select[Range[300], Function[n, # <= DivisorSigma[0, n] - # &@ DivisorSum[n, 1 &, CoprimeQ[#, n/#] &]]] (* or *)
Select[Range[300], Or[Count[#, p_ /; Last@ p >= 2] >= 2, Count[#, p_ /; Last@ p >= 3] == 1] &@ FactorInteger@ # &] (* Michael De Vlieger, Aug 01 2017 *)

is(n)=my(f=vecsort(factor(n)[,2],,4));#f && (f[1]>2 || (#f>1 && f[2]>1)) \\ Charles R Greathouse IV, Jun 07 2013

is(n)=factorback(factor(n)[,2]) > 2 \\ Charles R Greathouse IV, Aug 25 2016

More terms from James Sellers, Jun 20 2000

A317933 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Aug 12 2018

Multiplicative because A034444 is.
The first 2^20 terms are positive. Is the sequence nonnegative?
Records seem to be A001790, occurring at A000302 (apart from 4).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A001790, A034444, A317934 (denominators).

Cf. also A046643, A317831, A317925, A317937.

A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n,n,(A034444(n)-sumdiv(n,d,if((d>1)&&(dA317933perA317934(d)*A317933perA317934(n/d),0)))/2);
A317933(n) = numerator(A317933perA317934(n));

up_to = 65537;
\\ Faster:
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1+X)/(1-X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A034444(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A317933(k) / A317934(k) ~ sqrt(6)*n/Pi. - Vaclav Kotesovec, May 10 2025

cp's OEIS Frontend

A048109 Numbers having equally many squarefree and nonsquarefree divisors; number of unitary divisors of n (A034444) = number of non-unitary divisors of n (A048105).

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A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

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A064608 Partial sums of A034444: sum of number of unitary divisors from 1 to n.

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A322327 a(n) = A005361(n) * A034444(n).

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A048107 Numbers k such that the number of unitary divisors of k (A034444) is larger than the number of non-unitary divisors of k (A048105).

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A048111 Number of unitary divisors of n (A034444) < number of non-unitary divisors of n (A048105).

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