A065299 -id:A065299 - cp's OEIS frontend

A065300 Numbers k such that the sum of divisors of k is a squarefree number.

1, 2, 4, 5, 8, 9, 13, 16, 18, 20, 25, 26, 29, 36, 37, 41, 45, 49, 50, 61, 64, 72, 73, 74, 80, 100, 101, 104, 109, 113, 116, 117, 121, 122, 128, 137, 144, 146, 148, 157, 169, 173, 180, 181, 193, 196, 200, 208, 218, 225, 229, 234, 242, 244, 256, 257, 261, 277, 281

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Numbers k such that sigma(k) divides primorial(k), that is, A002110(k) mod A000203(k) = 0. - Gary Detlefs, May 02 2012

			For k = 100, sigma(100) = 217 = 7*31.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203 (sigma), A002110, A005117, A008683 (mu), A065299.

Select[Range@ 300, SquareFreeQ@ DivisorSigma[1, #] &] (* or *)
Select[Range@ 300, Abs@ MoebiusMu@ DivisorSigma[1, #] == 1 &] (* Michael De Vlieger, Mar 18 2017 *)

{ n=0; for (m = 1, 10^9, if (abs(moebius(sigma(m)))==1, write("b065300.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 15 2009

for(n=1, 300, if(issquarefree(sigma(n)), print1(n,", "))) \\ Indranil Ghosh, Mar 19 2017

from sympy import mobius, divisor_sigma
from sympy.ntheory.factor_ import core
[n for n in range(1,301) if abs(mobius(divisor_sigma(n, 1))) == 1] #* or *#
[n for n in range(1,301) if core(divisor_sigma(n,1)) == divisor_sigma(n,1)] # Indranil Ghosh, Mar 19 2017

Solutions to |mu(sigma(x))| = 1.

A065303 Neither n nor sigma(n) is squarefree.

Original entry on oeis.org

12, 24, 27, 28, 32, 40, 44, 48, 52, 54, 56, 60, 63, 68, 75, 76, 81, 84, 88, 90, 92, 96, 98, 99, 108, 112, 120, 124, 125, 126, 132, 135, 136, 140, 147, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 204, 207, 212, 216, 220

Offset: 1

Labos Elemer, Oct 29 2001

nonn

			n = 147 = 3*7*7, sigma(147) = 2*2*3*19 = 228.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000203, A008683, A065299, A065200, A065201, A065302.

Select[Range@ 220, Nor[SquareFreeQ@ #, SquareFreeQ@ DivisorSigma[1, #]] &] (* Michael De Vlieger, Mar 18 2017 *)
Select[Range[250],NoneTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2019 *)

n=0; for (m = 1, 10^9, if (!moebius(m) && !moebius(sigma(m)), write("b065303.txt", n++, " ", m); if (n==1000, return)) ) \\ Harry J. Smith, Oct 16 2009

sigmaSquarefree(f)=my(v=vector(#f~,i, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1))); for(i=2,#v, for(j=1,i-1, if(gcd(v[i],v[j])>1, return(0)))); for(i=1,#v, if(!issquarefree(v[i]), return(0))); 1
list(lim)=my(v=List()); forfactored(k=12,lim\1, if(!issquarefree(k) && !sigmaSquarefree(k[2]), listput(v,k[1]))); Vec(v) \\ Charles R Greathouse IV, Jan 08 2018

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def is_squarefree(n): return core(n) == n
print([i for i in range(1, 251) if not is_squarefree(i) and not is_squarefree(divisor_sigma(i,1))]) # Indranil Ghosh, Mar 18 2017

A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

Original entry on oeis.org

1, 2, 5, 13, 26, 29, 37, 41, 61, 73, 74, 101, 109, 113, 122, 137, 146, 157, 173, 181, 193, 218, 229, 257, 277, 281, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 541, 554, 569, 601, 613, 617, 626, 641, 653, 661, 673, 677

Offset: 1

Labos Elemer, Oct 29 2001

nonn

From Amiram Eldar, Mar 08 2025: (Start)
Number k such that A280710(k) * A280710(A000203(k)) = 1, or equivalently, A280710(k) * A280710(A048250(k)) = 1.
Squarefree numbers k whose prime factors are terms of A049097, and the elements of the set {p+1 , p|k} are pairwise coprime. (End)

			For k = 13, sigma(13) = 14 = 2*7 is squarefree.
For k = 26, sigma(26) = 1 + 2 + 13 + 26 = 42 = 2*3*7 is squarefree.
For k = 277 (prime), sigma(277) = 278 = 2*139 is squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000203, A008683, A048250, A065299, A065300.

Cf. A049097, A065303, A087248, A087249, A280710.

Select[Range[1000],AllTrue[{#,DivisorSigma[1,#]},SquareFreeQ]&] (* Harvey P. Dale, Aug 09 2014 *)

is(m) = abs(moebius(m))==1 && abs(moebius(sigma(m)))==1 \\ Harry J. Smith, Oct 15 2009

from sympy import divisor_sigma
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n)==n
print([n for n in range(1, 1001) if issquarefree(n) and issquarefree(divisor_sigma(n,1))]) # Indranil Ghosh, Mar 19 2017

A065302 Squarefree nonprime numbers whose sum of divisors is also squarefree.

Original entry on oeis.org

1, 26, 74, 122, 146, 218, 314, 362, 386, 458, 554, 626, 746, 794, 818, 842, 866, 914, 1082, 1202, 1226, 1322, 1346, 1418, 1466, 1514, 1538, 1658, 1706, 1754, 1874, 1994, 2018, 2042, 2066, 2138, 2186, 2234, 2258, 2306, 2402, 2426, 2474, 2594, 2642, 2762

Offset: 1

Labos Elemer, Oct 29 2001

nonn

All elements except the first, a(1)=1, are of the form 2*p, where p is a prime and p == 1 (mod 12). Also, sigma(2*p) = (1+2)*(1+p) = 6m where m = (1+p)/2 and m == 1 (mod 6). A squarefree composite number not of the form 2*p cannot be in the sequence since sigma is multiplicative. For example, sigma(p*q) = (1+p)*(1+q) is divisible by 4 for p,q > 2. - Walter Kehowski, Mar 21 2007

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000203, A008683, A065299, A065300, A065301, A065303.

Cf. A128607, A128608.

Select[Range[3000], !PrimeQ[#] && SquareFreeQ[#] && SquareFreeQ[DivisorSigma[1, #]] &] (* Amiram Eldar, Jun 05 2025 *)

isok(m) = !isprime(m) && moebius(m) && moebius(sigma(m)); \\ Harry J. Smith, Oct 16 2009

Name corrected by Amiram Eldar, Jun 05 2025

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A065300 Numbers k such that the sum of divisors of k is a squarefree number.

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A065303 Neither n nor sigma(n) is squarefree.

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A065301 Numbers k such that both k and the sum of its divisors are squarefree numbers.

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A065302 Squarefree nonprime numbers whose sum of divisors is also squarefree.

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