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

1,1

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].

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Christian Boyer, The smallest possible multiplicative magic squares.

Example

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.

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 *)

Crossrefs

Intersection of A000578 and A005101.

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

Sequence in context: A250137 A109399 A111029 * A177493 A162142 A233859

Adjacent sequences: A124578 A124579 A124580 * A124582 A124583 A124584

Keyword

nonn

Author

Tanya Khovanova, Dec 27 2006

Extensions

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

Status

approved