cp's OEIS Frontend

A350014 Numbers whose square has a number of divisors coprime to 6.

%I A350014 #18 Apr 06 2022 05:56:29
%S A350014 1,4,8,9,25,27,32,36,49,64,72,100,108,121,125,169,196,200,216,225,243,
%T A350014 256,288,289,343,361,392,441,484,500,512,529,576,675,676,729,800,841,
%U A350014 864,900,961,968,972,1000,1089,1125,1156,1225,1323,1331,1352,1369,1372,1444
%N A350014 Numbers whose square has a number of divisors coprime to 6.
%C A350014 a(n) = m in A001694 such that d(m^2) is not divisible by 3, where d(n) = A000005(n).
%C A350014 Supersequence of A051676 (composite numbers whose square has a prime number of divisors).
%C A350014 Subsequence of A001694 (powerful numbers).
%C A350014 Numbers whose prime factorization has only exponents that are congruent to {0, 2} mod 3 (A007494). - _Amiram Eldar_, Mar 31 2022
%H A350014 Michael De Vlieger, <a href="/A350014/b350014.txt">Table of n, a(n) for n = 1..10000</a>
%F A350014 a(n) = {m : gcd(d(m^2), 6) = 1}.
%F A350014 Sum_{n>=1} 1/a(n) = 15*zeta(3)/Pi^2 (= 10 * A240976). - _Amiram Eldar_, Mar 31 2022
%p A350014 A350014 := proc(n)
%p A350014     option remember ;
%p A350014     local a;
%p A350014     if n =1 then
%p A350014         1;
%p A350014     else
%p A350014         for a from procname(n-1)+1 do
%p A350014             if igcd(numtheory[tau](a^2),6) = 1 then
%p A350014                 return a;
%p A350014             end if;
%p A350014         end do:
%p A350014     end if;
%p A350014 end proc:
%p A350014 seq(A350014(n),n=1..20) ; # _R. J. Mathar_, Apr 06 2022
%t A350014 Select[Range[1500], CoprimeQ[DivisorSigma[0, #^2], 6] &] (* or *)
%t A350014 With[{nn = 1500}, Select[Union@ Flatten@ Table[a^2*b^3, {b, nn^(1/3)}, {a, Sqrt[nn/b^3]}], Mod[DivisorSigma[0, #^2], 3] != 0 &]]
%o A350014 (PARI) isok(m) = gcd(numdiv(m^2), 6) == 1; \\ _Michel Marcus_, Mar 04 2022
%Y A350014 Cf. A000005, A000290, A007494, A001694, A001651, A051676, A240976.
%K A350014 nonn,easy
%O A350014 1,2
%A A350014 _Michael De Vlieger_, Jan 17 2022