A048111 -id:A048111 - cp's OEIS frontend

A048146 Sum of non-unitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 6, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 24, 5, 0, 12, 16, 0, 0, 0, 30, 0, 0, 0, 41, 0, 0, 0, 36, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 36, 0, 48, 0, 0, 0, 48, 0, 0, 24, 62, 0, 0, 0, 36, 0, 0, 0, 105, 0, 0, 20, 40, 0, 0, 0, 84, 39, 0, 0, 64, 0, 0, 0, 72, 0, 54, 0

Offset: 1

Labos Elemer

nonn

			If n = 1000, the 12 non-unitary divisors are {2, 4, 5, 10, 20, 25, 40, 50, 100, 200, 250, 500} and their sum is a(n) = a(1000) = 1206. a(16) = a(2^4) = (2^4 - 2) / (2 - 1)= 14.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034444, A000203, A006881, A048105, A048106, A048107, A048109, A048111, A005117, A120944.

Cf. A002117, A072691.

us[n_Integer] := (d = Divisors[n]; l = Length[d]; k = 1; s = n; While[k < l, If[ GCD[ d[[k]], n/d[[k]] ] == 1, s = s + d[[k]]]; k++ ]; s); Table[ DivisorSigma[1, n] - us[n], {n, 1, 100} ]
(* Second program: *)
Table[DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 91}] (* Michael De Vlieger, Nov 20 2017 *)

a(n)=my(f=factor(n)); sigma(f)-prod(i=1, #f~, f[i, 1]^f[i, 2]+1) \\ Charles R Greathouse IV, Jun 17 2015

from sympy.ntheory.factor_ import divisor_sigma, udivisor_sigma
def A048146(n): return divisor_sigma(n)-udivisor_sigma(n) # Chai Wah Wu, Aug 22 2024

a(n) = A000203(n) - A034448(n) = sigma(n) - usigma(n). a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = (p^k - p) / (p - 1), for p = primes (A000040), pq = product of two distinct primes (A006881), pq...z = product of k (k >=2) distinct primes p, q, ..., z (A120944), p^k = prime powers (A000961(n) for n > 1) k = natural numbers (A000027).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * (1 - 1/zeta(3)) = 0.1382506... . - Amiram Eldar, Dec 09 2022

Edited by Jaroslav Krizek, Mar 01 2009

A082293 Numbers having exactly one square divisor > 1.

Original entry on oeis.org

4, 8, 9, 12, 18, 20, 24, 25, 27, 28, 40, 44, 45, 49, 50, 52, 54, 56, 60, 63, 68, 75, 76, 84, 88, 90, 92, 98, 99, 104, 116, 117, 120, 121, 124, 125, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 164, 168, 169, 171, 172, 175, 184, 188, 189, 198, 204, 207, 212

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

Numbers of the form m*p^2, p prime and m squarefree (A005117). [Corrected by Peter Munn, Nov 17 2020]

The asymptotic density of this sequence is (6/Pi^2)*Sum_{n>=1} 1/prime(n)^2 = 0.274933... (A222056). - Amiram Eldar, Jul 07 2020

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

Cf. A005117, A046951, A222056.
Complement of A048111 within A013929.
Subsequence of A252849.
Disjoint union of A048109 and A060687.
A285508 is a subsequence.

Select[Range[2, 200], MemberQ[{2, 3}, (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]]] && (Length[e] == 1 || e[[-2]] == 1) &] (* Amiram Eldar, Jul 07 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && f[1]>1 && f[1]<4 && (#f==1 || f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

from math import isqrt
from sympy import mobius, primerange
def A082293(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(g(x//p**2) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A046951(a(n)) = 2.

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)

A082294 Numbers having exactly two square divisors > 1.

Original entry on oeis.org

16, 32, 48, 80, 81, 96, 112, 160, 162, 176, 208, 224, 240, 243, 272, 304, 336, 352, 368, 405, 416, 464, 480, 486, 496, 528, 544, 560, 567, 592, 608, 624, 625, 656, 672, 688, 736, 752, 810, 816, 848, 880, 891, 912, 928, 944, 976, 992, 1040, 1053, 1056, 1072

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

Numbers of the form p^e * s where p is prime, e is 4 or 5 and s is squarefree and coprime to p. - David A. Corneth, Sep 01 2020

The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} (1/(p^3*(p+1)) + 1/(p^4*(p+1))) = 0.04680621631952059947... . - Amiram Eldar, Sep 25 2022

			81 has 3 square divisors: 1, 9 and 81, therefore 81 is a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A082295, A046951, A048111, A013929.

Select[Range[1000], MemberQ[{{4}, {5}}, Select[FactorInteger[#][[;;,2]], #1 > 1 &]] &] (* Amiram Eldar, Sep 01 2020 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); if(#f==1, f[1]>3&&f[1]<6, #f>1 && f[1]>3 && f[1]<6 && f[2]==1) \\ Charles R Greathouse IV, Oct 16 2015

A046951(a(n)) = 3.

A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 16, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -30, 48, 54, 48, 48, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -64, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 56, 84, -144, -68, -90, 96, -144, -72, -90

Offset: 1

Antti Karttunen, Nov 26 2024

sign

Arithmetic mean between A158523 and A178450.

Apparently differs from A378433 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433, A378435 (Dirichlet inverse).

A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);

a(n) = (1/2) * (A158523(n)+A178450(n)).

A082295 Numbers having more than two square divisors > 1.

Original entry on oeis.org

36, 64, 72, 100, 108, 128, 144, 180, 192, 196, 200, 216, 225, 252, 256, 288, 300, 320, 324, 360, 384, 392, 396, 400, 432, 441, 448, 450, 468, 484, 500, 504, 512, 540, 576, 588, 600, 612, 640, 648, 675, 676, 684, 700, 704, 720, 729, 756, 768, 784, 792, 800

Offset: 1

Reinhard Zumkeller, Apr 08 2003

nonn

If n is in the sequence, so is m*n. - Charles R Greathouse IV, Oct 16 2015

The asymptotic density of this sequence is 1 - (6/Pi^2) * (1 + Sum_{p prime} (1/p^2 + 1/(p^3*(p+1)) + 1/(p^4*(p+1)))) = 0.07033321843992718294... . - Amiram Eldar, Sep 25 2022

			n=200 has 4 square divisors: 1, 4, 25 and 100, therefore 200 is a term.

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

Cf. A082294, A048111, A046951.

Select[Range[1000],Length[Rest[Select[ Divisors[#], IntegerQ[ Sqrt[ #]]&]]]> 2&] (* Harvey P. Dale, Jan 08 2014 *)

is(n)=my(f=vecsort(factor(n)[,2],,4)); #f && (f[1]>5 || (#f>1 && f[2]>1)) \\ Charles R Greathouse IV, Oct 16 2015

A046951(a(n)) > 3.

a(n) < 17n for n > 25. - Charles R Greathouse IV, Oct 16 2015

A378433 Dirichlet inverse of A325973, where A325973 is the arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, -3, -4, 5, -6, 12, -8, -9, 9, 18, -12, -20, -14, 24, 24, 15, -18, -27, -20, -30, 32, 36, -24, 36, 20, 42, -24, -40, -30, -72, -32, -27, 48, 54, 48, 42, -38, 60, 56, 54, -42, -96, -44, -60, -54, 72, -48, -60, 35, -60, 72, -70, -54, 72, 72, 72, 80, 90, -60, 120, -62, 96, -72, 45, 84, -144, -68, -90, 96, -144, -72, -72

Offset: 1

Antti Karttunen, Nov 26 2024

sign

Apparently differs from A378434 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A034448, A048111, A048250, A325973, A378434.

A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
memoA378433 = Map();
A378433(n) = if(1==n,1,my(v); if(mapisdefined(memoA378433,n,&v), v, v = -sumdiv(n,d,if(dA325973(n/d)*A378433(d),0)); mapput(memoA378433,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA325973(n/d) * a(d).

A273785 Numbers n where a composite c < n exists such that n^(c-1) == 1 (mod c^2), i.e., such that c is a "base-n Wieferich pseudoprime".

Original entry on oeis.org

17, 26, 33, 37, 49, 65, 73, 80, 81, 82, 97, 99, 101, 109, 113, 129, 145, 146, 161, 163, 168, 170, 177, 181, 182, 193, 197, 199, 201, 209, 217, 224, 225, 226, 239, 241, 242, 244, 251, 253, 257, 268, 273, 289, 293, 301, 305, 321, 323, 325, 337, 353, 360, 361

Offset: 1

Felix Fröhlich, May 30 2016

nonn

Contains n+1 for n in A048111. - Robert Israel, Apr 20 2017

			15 satisfies the congruence 26^(15-1) == 1 (mod 15^2) and 15 < 26, so 26 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A048111, A240719, A244752, A255885, A256517, A267288, A268310, A273339, A273340.

N:= 1000: # to get all terms <= N
Res:= {}:
for c from 4 to N-1 do
  if not isprime(c) then
    for m in map(rhs@op, [msolve(x^(c-1)-1, c^2)]) do
       if m > c and m <= N then Res:= Res union {m, seq(k*c^2+m, k=1..(N-m)/c^2)}
       else Res:= Res union {seq(k*c^2+m, k=1..(N-m)/c^2)}
       fi
    od
  fi
od:
sort(convert(Res,list)); # Robert Israel, Apr 20 2017

nn = 361; c = Select[Range@ nn, CompositeQ]; Select[Range@ nn, Function[n, Count[TakeWhile[c, # <= n &], k_ /; Mod[n^(k - 1), k^2] == 1] > 0]] (* Michael De Vlieger, May 30 2016 *)

is(n) = forcomposite(c=1, n-1, if(Mod(n, c^2)^(c-1)==1, return(1))); return(0)

A378435 Dirichlet inverse of the arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Original entry on oeis.org

1, 3, 4, 4, 6, 12, 8, 6, 7, 18, 12, 16, 14, 24, 24, 9, 18, 21, 20, 24, 32, 36, 24, 24, 16, 42, 16, 32, 30, 72, 32, 15, 48, 54, 48, 25, 38, 60, 56, 36, 42, 96, 44, 48, 42, 72, 48, 36, 29, 48, 72, 56, 54, 48, 72, 48, 80, 90, 60, 96, 62, 96, 56, 24, 84, 144, 68, 72, 96, 144, 72, 33, 74, 114, 64, 80, 96, 168, 80, 54, 34

Offset: 1

Antti Karttunen, Nov 26 2024

sign

The first negative term is a(2592) = -48.

Apparently differs from A325973 at positions given by A048111: 16, 32, 36, 48, 64, 72, 80, 81, 96, ...

Antti Karttunen, Table of n, a(n) for n = 1..20000

Dirichlet inverse of A378434.

Cf. A034448, A048111, A048250, A158523, A178450, A325973, A378433.

A158523(n) = { my(f = factor(n)); prod(i = 1, #f~, (-1)^f[i, 2]*(f[i, 1]+1)*f[i, 1]^(f[i, 2]-1)); }; \\ From A158523
A178450(n) = { my(f=factor(n)); prod(i=1, #f~, if(!(f[i,2]%2), 2*(f[i, 1]^(f[i, 2]/2)), -(1+f[i,1])*(f[i, 1]^((f[i, 2]-1)/2)))); };
A378434(n) = ((A158523(n)+A178450(n))/2);
memoA378435 = Map();
A378435(n) = if(1==n,1,my(v); if(mapisdefined(memoA378435,n,&v), v, v = -sumdiv(n,d,if(dA378434(n/d)*A378435(d),0)); mapput(memoA378435,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA378434(n/d) * a(d).

A048195 Numbers k for which binomial(k, floor(k/2)) has fewer unitary than non-unitary divisors.

Original entry on oeis.org

10, 25, 26, 27, 28, 29, 30, 34, 36, 37, 38, 40, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 58, 60, 61, 62, 63, 64, 66, 68, 69, 70, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 108, 109, 110

Offset: 1

Labos Elemer

nonn

A048111 applied to central binomial coefficients.

			k = 58: binomial(58,29) has 20480 divisors, 8192 unitary ones and 12288 non-unitary ones, and 8192 < 12288.

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

Cf. A001405, A034444, A048105, A048111.

q[n_] := Module[{e = FactorInteger[Binomial[n, Floor[n/2]]][[;; , 2]]}, Times @@ (e + 1) > 2^(Length[e] + 1)]; Select[Range[120], q] (* Amiram Eldar, Oct 05 2024 *)

nbud(n) = 1<A034444
isok(n) = my(b=binomial(n, n\2)); numdiv(b) > 2*nbud(b); \\ Michel Marcus, Mar 15 2018

A034444(A001405(k)) < A048105(A001405(k)).

More terms from Michel Marcus, Mar 15 2018

cp's OEIS Frontend

A048146 Sum of non-unitary divisors of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A082293 Numbers having exactly one square divisor > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A082294 Numbers having exactly two square divisors > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378434 Arithmetic mean between the Dirichlet inverses of {sum of unitary divisors} and {sum of squarefree divisors}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A082295 Numbers having more than two square divisors > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378433 Dirichlet inverse of A325973, where A325973 is the arithmetic mean of {sum of unitary divisors} and {sum of squarefree divisors}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI