A065303 -id:A065303 - cp's OEIS frontend

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

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

A087249 Squarefree numbers k such that sigma(k) is not squarefree.

Original entry on oeis.org

3, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 38, 39, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 62, 65, 66, 67, 69, 70, 71, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 102, 103, 105, 106, 107, 110, 111, 114, 115, 118, 119, 123, 127, 129, 130

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			For k=7: sigma(7) = 2*2*2 = 8.
For k=10: sigma(10) = 1 + 2 + 5 + 10 = 18 = 2*3*3.

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

Complement of A065301 within A005117.

Cf. A065303, A087248.

Select[Range[150],SquareFreeQ[#]&&!SquareFreeQ[DivisorSigma[1,#]]&]  (* Harvey P. Dale, Feb 06 2011 *)

is(k) = issquarefree(k) && !issquarefree(sigma(k)); \\ Amiram Eldar, Jun 15 2024

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

A086368 Nonsquarefree numbers k such that sigma(k) is squarefree.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 25, 36, 45, 49, 50, 64, 72, 80, 100, 104, 116, 117, 121, 128, 144, 148, 169, 180, 196, 200, 208, 225, 234, 242, 244, 256, 261, 289, 292, 296, 320, 325, 333, 361, 369, 404, 436, 441, 450, 452, 464, 488, 512, 529, 548, 549, 576, 578, 584, 592

Offset: 1

Labos Elemer, Sep 05 2003

nonn

			k = 45 = 3*3*5 and sigma(45) = 78 = 2*3*13.

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

Intersection of A013929 and A065300.

Cf. A000203 (sigma), A005117, A065303, A087249, A065301.

filter:= n -> not numtheory:-issqrfree(n) and numtheory:-issqrfree(numtheory:-sigma(n)):
select(filter, [$1..1000]); # Robert Israel, May 11 2018

q[k_] := ! SquareFreeQ[k] && SquareFreeQ[DivisorSigma[1, k]]; Select[Range[600], q] (* Amiram Eldar, Feb 24 2025 *)

isok(k) = !issquarefree(k) && issquarefree(sigma(k)); \\ Amiram Eldar, Feb 24 2025

A283800 Numbers such that the sum of trits of its balanced ternary representation is 1 or -1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 29, 33, 35, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 69, 71, 73, 75, 77, 79, 81, 83, 87, 89, 95, 97, 99, 101, 105, 107, 113, 127, 129, 133, 135, 137, 139, 141, 143, 145, 147, 151, 153, 155, 157, 159, 161

Offset: 1

Lei Zhou, Mar 16 2017

			3 = 10 in balanced ternary (bt) notation, 1+0 = 1, so 3 is in the list;
...
11 = 11T in bt notation, 1+1+T = 1, here T represent -1, so 11 is in the list;
13 = 111 in bt notation, 1+1+1 = 3, so 13 is NOT in the list.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A065303, A174658.

BTDigits[m_Integer,
   g_] :=(*This is to determine digits of a number in balanced \
ternary notation.*)
  Module[{n = m, d, sign, t = g},
   If[n != 0, If[n > 0, sign = 1, sign = -1; n = -n];
    d = Ceiling[Log[3, n]]; If[3^d - n <= ((3^d - 1)/2), d++];
    While[Length[t] < d, PrependTo[t, 0]];
    t[[Length[t] + 1 - d]] = sign;
    t = BTDigits[sign*(n - 3^(d - 1)), t]]; t];
n = 0; Table[While[n++; g = {}; bt = BTDigits[n, g]; s = Total[bt];
  Abs[s] != 1]; n, {i, 1, 61}]

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

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A087249 Squarefree numbers k such that sigma(k) is not squarefree.

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

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A086368 Nonsquarefree numbers k such that sigma(k) is squarefree.

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A283800 Numbers such that the sum of trits of its balanced ternary representation is 1 or -1.

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Mathematica