A068140 Smaller of two consecutive numbers each divisible by a cube greater than one.

80, 135, 296, 343, 351, 375, 512, 567, 624, 728, 783, 944, 999, 1160, 1215, 1375, 1376, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2079, 2240, 2295, 2375, 2400, 2456, 2511, 2624, 2672, 2727, 2888, 2943, 3087, 3104, 3159, 3320, 3375, 3429, 3536, 3591

Offset: 1

Author

Amarnath Murthy, Feb 22 2002

Keywords

• easy
• nonn

Comments

Cubeful numbers with cubeful successors. This is to cubes as A068781 is to squares. 1375 is the smallest of three consecutive numbers divisible by a cube, since 1375 = 5^3 * 11 and 1376 = 2^5 * 43 and 1377 = 3^4 * 17. What is the smallest of four consecutive numbers divisible by a cube? Of n consecutive numbers divisible by a cube? - Jonathan Vos Post, Sep 18 2007

22624 is the smallest of four consecutive numbers each divisible by a cube, with factorizations 2^5 * 7 * 101, 5^3 * 181, 2 * 3^3 * 419, and 11^3 * 17. - D. S. McNeil, Dec 10 2010

18035622 is the smallest of five consecutive numbers each divisible by a cube. 4379776620 is the smallest of six consecutive numbers each divisible by a cube. 1204244328624 is the smallest of seven consecutive numbers each divisible by a cube. - Donovan Johnson, Dec 13 2010

The sequence is the union, over all pairs of distinct primes (p,q), of numbers == 0 mod p^3 and == -1 mod q^3 or vice versa. - Robert Israel, Aug 13 2018

The asymptotic density of this sequence is 1 - 2/zeta(3) + Product_{p prime} (1 - 2/p^3) = 1 - 2 * A088453 + A340153 = 0.013077991848467056243... - Amiram Eldar, Feb 16 2021

Examples

			343 is a term as 343 = 7^3 and 344= 2^3 * 43.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A046099, A063528, A068781, A068782, A068783, A068784, A088453, A122692, A174113, A340152, A340153.

Programs

Maple

isA068140 := proc(n)
    isA046099(n) and isA046099(n+1) ;
end proc:
for n from 1 to 4000 do
    if isA068140(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 08 2015

Mathematica

a = b = 0; Do[b = Max[ Transpose[ FactorInteger[n]] [[2]]]; If[a > 2 && b > 2, Print[n - 1]]; a = b, {n, 2, 5000}]
Select[Range[2, 6000], Max[Transpose[FactorInteger[ # ]][[2]]] > 2 && Max[Transpose[FactorInteger[ # + 1]][[2]]] > 2 &] (* Jonathan Vos Post, Sep 18 2007 *)
SequencePosition[Table[If[AnyTrue[Rest[Divisors[n]],IntegerQ[Surd[#,3]]&],1,0],{n,3600}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 18 2020 *)

Formula

{k such that k is in A046099 and k+1 is in A046099}. - Jonathan Vos Post, Sep 18 2007

Extensions

Edited and extended by Robert G. Wilson v, Mar 02 2002

Title edited, cross-references added by Matthew Vandermast, Dec 09 2010

Definition clarified by Harvey P. Dale, Apr 18 2020