A080402 -id:A080402 - cp's OEIS frontend

A080401 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is squarefree.

1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 22, 23, 25, 29, 31, 32, 37, 38, 40, 44, 47, 48, 49, 50, 52, 53, 58, 59, 61, 62, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 83, 88, 89, 92, 97, 98, 99, 101, 103, 109, 113, 116, 117, 118, 121, 122, 124, 127, 128, 131, 137

Offset: 1

Labos Elemer, Mar 19 2003

nonn

If m*k is in the sequence with m and k coprime, then m and k must be in the sequence. - Robert Israel, Mar 29 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A001157, A005117, A065300, A080402 (complement).

select(n -> numtheory:-issqrfree(numtheory:-sigma[2](n)), [$1..1000]); # Robert Israel, Mar 29 2019

Do[s=MoebiusMu[DivisorSigma[2, n]]; If[ !Equal[s, 0], Print[n]], {n, 1, 1000}]
Select[Range[200],SquareFreeQ[DivisorSigma[2,#]]&] (* Harvey P. Dale, Jun 17 2014 *)

isok(n) = issquarefree(sigma(n, 2)); \\ Michel Marcus, Mar 29 2019

abs(mu(sigma_2(a(n)))) = 1.

A080403 Numbers n such that both A000203(n) and A001157(n) are squarefree: sum of divisors and squares of divisors of n are squarefree.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 13, 16, 18, 20, 25, 29, 37, 49, 50, 61, 64, 72, 73, 80, 101, 109, 113, 116, 117, 121, 122, 128, 137, 144, 148, 169, 173, 181, 200, 208, 218, 229, 242, 244, 261, 277, 281, 289, 292, 313, 317, 320, 325, 333, 353, 362, 373, 389, 397, 401, 404, 409

Offset: 1

Labos Elemer, Mar 19 2003

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

Cf. A001157, A000203, A065300, A080401, A080402.

Do[s1=MoebiusMu[DivisorSigma[1, n]]; s2=MoebiusMu[DivisorSigma[2, n]]; If[ !Equal[s1, 0]&&!Equal[s2, 0], Print[n]], {n, 1, 1000}]

cp's OEIS Frontend

A080401 Numbers k such that the sum of the squares of the divisors of k (A001157(k)) is squarefree.

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