%I #29 Sep 08 2022 08:45:05

%S 432,576,648,1600,2000,2160,2880,2916,3024,3136,3240,4032,4536,4752,

%T 4800,5000,5488,5616,6000,6336,7128,7344,7488,7744,8208,8424,9408,

%U 9792,9936,10125,10800,10816,10944,11016,11200,12312,12528,13248,13392

%N Numbers k such that gcd(d(k^3), d(k)) is not a power of 2.

%C The complement of this sequence in the positive integers A000027 is A069782. - _M. F. Hasler_, Jan 18 2015

%C The numbers of the form 4*3^(7*m - 1), m >= 1, are terms. - _Marius A. Burtea_, Oct 18 2019

%H Charles R Greathouse IV, <a href="/A069781/b069781.txt">Table of n, a(n) for n = 1..10000</a>

%F log_2(gcd(A000005(n^3), A000005(n))) is nonintegral.

%e For n<100000, gcd[d(n^3),d[n]] = {5,7,10,14,20,28,40,80} which is obtained for n={20736,576,432,2880,54000,20160,2160,15120} respectively.

%t f[x_] := GCD[DivisorSigma[0, x^3], DivisorSigma[0, x]] Do[s=f[n]; If[ !IntegerQ[Log[2, s]], Print[n]], {n, 1, 100000}]

%t Select[Range[14000],!IntegerQ[Log[2,GCD[DivisorSigma[0,#^3], DivisorSigma[ 0,#]]]]&] (* _Harvey P. Dale_, Mar 20 2018 *)

%o (PARI) is(n)=my(f=factor(n)[,2], g=gcd(prod(i=1,#f,3*f[i]+1), prod(i=1,#f,f[i]+1))); g!=1<<valuation(g, 2) \\ _Charles R Greathouse IV_, Oct 16 2015

%o (Magma) f:=func<n| Gcd(#Divisors(n^3),#Divisors(n))>; [k:k in [1..14000]| not IsIntegral(Log(2,f(k)))]; // _Marius A. Burtea_, Oct 18 2019

%Y Cf. A000005, A061701, A037992, A069780, A069782.

%K nonn

%O 1,1

%A _Labos Elemer_, Apr 08 2002