cp's OEIS Frontend

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

%I A361634 #25 Apr 08 2023 23:26:16
%S A361634 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,
%T A361634 34,35,36,37,38,39,41,42,43,46,47,48,49,51,53,55,57,58,59,61,62,64,65,
%U A361634 66,67,69,70,71,73,74,77,78,79,80,81,82,83,85,86,87,89,91,93,94
%N A361634 Integers whose number of square divisors is coprime to the number of their nonsquare divisors.
%C A361634 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.
%H A361634 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%e A361634 48 has 3 square divisors (1, 4, and 16) and 7 nonsquare ones. Consequently, gcd(3,7)=1, thus 48 is a term.
%t A361634 Select[Range[100],CoprimeQ[Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#]),DivisorSigma[0,#]-Total@(Boole/@IntegerQ/@Sqrt/@Divisors[#])]&]
%o A361634 (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, ", "))
%o A361634 (PARI) isok(n) = gcd(numdiv(n), numdiv(sqrtint(n/core(n))))==1 \\ _Andrew Howroyd_, Mar 19 2023
%Y A361634 Cf. A210490, A005117 (subsequences), A046951 (number of square divisors), A056595 (number of nonsquare divisors).
%K A361634 nonn
%O A361634 1,2
%A A361634 _Waldemar Puszkarz_, Mar 19 2023