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.

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2046, 4095, 8190, 16380, 32760, 65520, 131040, 262080, 524160, 1048320, 2097144, 4193280, 8386560, 16773900, 33547800, 67095600, 134191200, 268382400, 536215680, 1073709000, 2144142000, 4288284000, 8527559040, 16908091200, 27935107200

Offset: 1

Author

Labos Elemer, Aug 16 2000

Keywords

• nonn
• fini
• full

Examples

			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.
		

Crossrefs

Cf. A000005, A029837, A034884, A035033, A035034, A035035, A056757-A056767, A056781.

Programs

Mathematica

Table[Last@ Select[Range @@ (2^{n - 1, n}), DivisorSigma[0, #]^3 > # &], {n, 22}] (* Michael De Vlieger, Dec 31 2016 *)

PARI

a(n) = {k = 2^n; while(numdiv(k)^3 <= k, k--); k;} \\ Michel Marcus, Dec 11 2013

Formula

Largest terms of A056757 between 2^(n-1) and 2^n.

Extensions

a(32) from Michel Marcus, Dec 11 2013

a(33)-a(35) and keyword "full" added by Amiram Eldar, Feb 23 2025