A124581 Abundant cubes.

216, 1000, 1728, 2744, 5832, 8000, 10648, 13824, 17576, 21952, 27000, 46656, 64000, 74088, 85184, 110592, 125000, 140608, 157464, 175616, 216000, 287496, 314432, 343000, 373248, 438976, 474552, 512000, 592704, 681472, 729000, 778688, 884736

Offset: 1

Author

Tanya Khovanova, Dec 27 2006

Keywords

• nonn

Comments

Abundant cubes can't be prime powers for obvious reasons. Hence all these numbers can be represented as a^3*b^3 for some coprime a and b. a^3*b^3 is the magic product of the following magic 3 X 3 multiplicative square: [a*b^2, 1, a^2*b; a^2, ab, b^2; b, a^2*b^2; a].

Examples

			216 is in the sequence because 216=6^3 and the sum of the proper divisors of 216 is 108+72+54+...+3+2+1 > 216.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Christian Boyer, The smallest possible multiplicative magic squares.

Crossrefs

Intersection of A000578 and A005101.

Cf. A111029 = magic products of 3 X 3 multiplicative magic squares.

Programs

Maple

isA005101 := proc(n) if numtheory[sigma](n) > 2*n then RETURN(true) ; else RETURN(false) ; fi ; end : for n from 1 to 120 do if isA005101(n^3) then printf("%d,",n^3) ; fi ; od ; # R. J. Mathar, Jan 07 2007
with(numtheory): a:=proc(n) if sigma(n^3)>2*n^3 then n^3 else fi end: seq(a(n),n=1..110); # Emeric Deutsch, Jan 10 2007

Mathematica

Select[Range[100]^3, DivisorSigma[1, #] > 2# &] (* Amiram Eldar, Aug 14 2019 *)

Extensions

More terms from R. J. Mathar and Emeric Deutsch, Jan 07 2007