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A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.

%I A056767 #19 Feb 23 2025 02:40:14
%S A056767 2,4,8,16,32,64,128,256,512,1024,2046,4095,8190,16380,32760,65520,
%T A056767 131040,262080,524160,1048320,2097144,4193280,8386560,16773900,
%U A056767 33547800,67095600,134191200,268382400,536215680,1073709000,2144142000,4288284000,8527559040,16908091200,27935107200
%N A056767 Largest number of binary size n (i.e., between (n-1)-th and n-th powers of 2) with the following property: cube of its number of divisors is larger than the number itself.
%F A056767 Largest terms of A056757 between 2^(n-1) and 2^n.
%e A056767 These maximal terms are usually "near" to 2^n. For n=1..10 they are equal to 2^n. At n=21, a(21)=2097144, 1048576 < a(21) < 2097144 = 8*27*7*19*73 has d=128 divisors, of which the cube is d^3d=2097152. So this maximum is near to but still less than d^3.
%t A056767 Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* _Michael De Vlieger_, Dec 31 2016 *)
%o A056767 (PARI) a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ _Michel Marcus_, Dec 11 2013
%Y A056767 Cf. A000005, A029837, A034884, A035033, A035034, A035035, A056757-A056767, A056781.
%K A056767 nonn,fini,full
%O A056767 1,1
%A A056767 _Labos Elemer_, Aug 16 2000
%E A056767 a(32) from _Michel Marcus_, Dec 11 2013
%E A056767 a(33)-a(35) and keyword "full" added by _Amiram Eldar_, Feb 23 2025