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

Author

Waldemar Puszkarz, Mar 19 2023

Keywords

• nonn

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.

Examples

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

Links

• Index entries for sequences related to divisors of numbers

Crossrefs

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

Programs

Mathematica

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

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