A048105 -id:A048105 - 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

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))).

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)

A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

Original entry on oeis.org

1, 2, 5, 6, 10, 13, 14, 21, 22, 27, 29, 30, 33, 34, 37, 38, 41, 42, 44, 46, 57, 58, 61, 65, 66, 69, 70, 73, 75, 77, 78, 82, 85, 86, 93, 94, 98, 101, 102, 105, 106, 109, 110, 113, 114, 116, 118, 122, 124, 129, 130, 133, 135, 137, 138, 141, 142, 145, 147, 154, 157

Offset: 1

Amiram Eldar, May 14 2021

nonn

			1 is a term since A048105(1) = A048105(2) = 0.
27 is a term since A048105(27) = A048105(28) = 2.

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

Cf. A048105, A064115, A335328, A344315.

Similar sequences: A005237, A006049, A343819, A344312, A344313.

nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[200], nd[#] == nd[# + 1] &]

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

A336316 The number of non-unitary divisors in the conjugated prime factorization of n: a(n) = A048105(A122111(n)).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 4, 2, 0, 6, 0, 7, 2, 5, 6, 8, 0, 2, 8, 1, 4, 9, 0, 10, 0, 8, 10, 4, 0, 11, 12, 11, 2, 12, 4, 13, 6, 2, 14, 14, 0, 3, 2, 14, 8, 15, 0, 8, 4, 17, 16, 16, 0, 17, 18, 5, 0, 12, 8, 18, 10, 20, 4, 19, 0, 20, 20, 2, 12, 6, 12, 21, 2, 1, 22, 22, 4, 16, 24, 23, 6, 23, 0, 11, 14, 26, 26, 20, 0, 24

Offset: 1

Antti Karttunen, Jul 18 2020

nonn

Equally, the number of divisors in the conjugated prime factorization of n minus the number of its unitary divisors.

Note that A001221(A122111(n)) = A001221(n) for all n.

Antti Karttunen, Table of n, a(n) for n = 1..12000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A000005, A001221, A034444, A048105, A122111, A286621, A336315.

Cf. A055932 (the positions of zeros).

A336315(n) = if(1==n,n,my(p=apply(primepi,factor(n)[,1]~),m=1+p[1]); for(i=2, #p, m *= (1+p[i]-p[i-1])); (m));
A336316(n) = (A336315(n)-(2^omega(n)));

a(n) = A336315(n) - A034444(n) = A000005(A122111(n)) - 2^A001221(n).

a(n) = A048105(A122111(n)).

A344315 a(n) is the least number k such that A048105(k) = A048105(k+1) = 2*n, and 0 if it does not exist.

Original entry on oeis.org

1, 27, 135, 2511, 2295, 6975, 5264, 12393728, 12375, 2200933376, 108224, 257499, 170624, 3684603215871, 4402431, 2035980763136, 126224, 41680575, 701443071, 46977524, 1245375, 2707370000, 4388175, 3129761024, 1890944

Offset: 0

Amiram Eldar, May 14 2021

There are no two consecutive numbers with an odd number of non-unitary divisors, since A048105(k) is odd only if k is a perfect square.

a(25) <= 1965640805422351777791, a(26) <= 3127059999. In general, a(n) <= A215199(n+1). - Daniel Suteu, May 20 2021

			a(0) = 1 since A048105(1) = A048105(2) = 0.
a(1) = 27 since A048105(27) = A048105(28) = 2.

Cf. A048105, A075036, A093548, A309181, A344314.

nd[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[max_] := Module[{s = Table[0, {max}], k = 2, c = 0, nd1 = 0}, While[c < max, If[(nd2 = nd[k]) == nd1 && nd2 < 2*max && s[[nd2/2 + 1]] == 0, c++; s[[nd2/2 + 1]] = k - 1]; nd1 = nd2; k++]; s]; seq[7]

A048105(n) = numdiv(n) - 2^omega(n);
isok(n,k) = A048105(k) == 2*n && A048105(k+1) == 2*n;
a(n) = for(k=1, oo, if(isok(n, k), return(k))); \\ Daniel Suteu, May 16 2021

a(13)-a(24) confirmed by Martin Ehrenstein, May 20 2021

A344732 Numbers k such that k divides Sum_{j=1..k} A048105(j).

Original entry on oeis.org

1, 2, 3, 54, 58, 62, 71, 10535, 10541, 10579, 135242, 135243, 1733777, 1733781, 1733895, 1733905, 1733999, 22216757, 22216765, 22216790, 22216808, 22216814, 46745561148, 46745561156

Offset: 1

Amiram Eldar, May 27 2021

The corresponding quotients Sum_{j=1..k} A048105(j)/k are 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 9, 9, ...

a(25) > 7.5*10^10, if it exists.

			a(1) = 1 since A048105(1) = 0 is divisible by 1.
a(4) = 54 since Sum_{j=1..54} A048105(j) = 54 is divisible by 54.

Cf. A048105.
The non-unitary version of A050226.
Similar sequences: A064610, A344731, A344733.

s[1] = 0; s[n_] := s[n] = s[n - 1] + DivisorSigma[0, n] - 2^PrimeNu[n]; Select[Range[140000], Divisible[s[#], #] &]

A062171 Number of non-unitary divisors of n (A048105) > number of distinct prime divisors of n (A001221).

Original entry on oeis.org

8, 16, 24, 27, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 104, 108, 112, 120, 125, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 216, 220, 224, 225, 228, 232, 234, 240

Offset: 1

Jason Earls, Aug 28 2001

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A048105, A001221, A048108.

nud(n) = numdiv(n)-2^omega(n); j=[]; for(n=1,500, if(nud(n)>omega(n),j=concat(j,n))); j

is(n)=my(f=factor(n)[,2]); prod(i=1,#f,f[i]+1)>2^#f+#f \\ Charles R Greathouse IV, Sep 18 2015

Offset corrected by Charles R Greathouse IV, Sep 18 2015

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

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A344314 Number k such that k and k+1 have the same number of nonunitary divisors (A048105).

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

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Maple

Mathematica

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PARI

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A336316 The number of non-unitary divisors in the conjugated prime factorization of n: a(n) = A048105(A122111(n)).

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