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A068140 Smaller of two consecutive numbers each divisible by a cube greater than one. 21

%I #31 Feb 16 2021 04:29:39

%S 80,135,296,343,351,375,512,567,624,728,783,944,999,1160,1215,1375,

%T 1376,1431,1592,1624,1647,1808,1863,2024,2079,2240,2295,2375,2400,

%U 2456,2511,2624,2672,2727,2888,2943,3087,3104,3159,3320,3375,3429,3536,3591

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

%C 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

%C 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

%C 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

%C 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

%C 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

%H Robert Israel, <a href="/A068140/b068140.txt">Table of n, a(n) for n = 1..10000</a>

%F {k such that k is in A046099 and k+1 is in A046099}. - _Jonathan Vos Post_, Sep 18 2007

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

%p isA068140 := proc(n)

%p isA046099(n) and isA046099(n+1) ;

%p end proc:

%p for n from 1 to 4000 do

%p if isA068140(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Dec 08 2015

%t a = b = 0; Do[b = Max[ Transpose[ FactorInteger[n]] [[2]]]; If[a > 2 && b > 2, Print[n - 1]]; a = b, {n, 2, 5000}]

%t Select[Range[2, 6000], Max[Transpose[FactorInteger[ # ]][[2]]] > 2 && Max[Transpose[FactorInteger[ # + 1]][[2]]] > 2 &] (* _Jonathan Vos Post_, Sep 18 2007 *)

%t 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 *)

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

%K easy,nonn

%O 1,1

%A _Amarnath Murthy_, Feb 22 2002

%E Edited and extended by _Robert G. Wilson v_, Mar 02 2002

%E Title edited, cross-references added by _Matthew Vandermast_, Dec 09 2010

%E Definition clarified by _Harvey P. Dale_, Apr 18 2020

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)