A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94

Offset

1,2

Comments

Appears to be a supersequence of A210490, and so also of positive squares and squarefree numbers (A005117). The first term that belongs in here but not in A210490 is 48. The nonsquarefree terms that are not squares are of the form p^(4k)*a, where a is a squarefree number, p is prime, and k > 0. About half of perfect numbers are of this form; one example is 496 = 2^4*31. The sequence has an asymptotic density of about 0.6420.

Links

Table of n, a(n) for n=1..69.

Index entries for sequences related to divisors of numbers

Example

48 has 3 square divisors (1, 4, and 16) and 7 nonsquare ones. Consequently, gcd(3,7)=1, thus 48 is a term.

Mathematica

Select[Range[100], CoprimeQ[Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#]), DivisorSigma[0, #]-Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#])]&]

Prog

(PARI) for(n=1, 100, a=divisors(n); c=0; for(i=1, #a, issquare(a[i])&&c++); gcd(#a-c, c)==1&&print1(n, ", "))
(PARI) isok(n) = gcd(numdiv(n), numdiv(sqrtint(n/core(n))))==1 \\ Andrew Howroyd, Mar 19 2023

Crossrefs

Cf. A210490, A005117 (subsequences), A046951 (number of square divisors), A056595 (number of nonsquare divisors).

Sequence in context: A039217 A239289 A131511 * A210490 A340682 A166155

Adjacent sequences: A361631 A361632 A361633 * A361635 A361636 A361637

Keyword

nonn

Author

Waldemar Puszkarz, Mar 19 2023

Status

approved