A174113 -id:A174113 - cp's OEIS frontend

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

Amarnath Murthy, Feb 22 2002

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

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

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

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

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

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 *)

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

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

A330480 Earliest start of a run of n numbers divisible by a fourth power larger than one.

Original entry on oeis.org

16, 80, 33614, 202099373, 40280549372, 430995495889374, 77405340617896874

Offset: 1

Jud McCranie, Dec 16 2019

De Koninck's book says that "most likely" a(7)=77405340617896874.

			33614 is divisible by 7^4, 33615 is divisible by 3^4, and 33616 is divisible by 2^4.  This is the smallest number with this property, so a(3)=33614.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A088080, A174113.

Cf. A045882, A271443, A330481, A330482, A330486, A330483, A330484, A330485.

a(7) from Giovanni Resta, Dec 18 2019

A330482 Earliest start of a run of n numbers divisible by a sixth power larger than one.

Original entry on oeis.org

64, 16767, 26890623, 1507545109375, 777562026420218750, 283435321166212288109372

Offset: 1

Jud McCranie, Dec 16 2019

De Konnick's book gives probable terms a(5)=777562026420218750 and a(6)=283435321166212288109372.

			26890623 is divisible by 3^6, 26890624 is divisible by 2^6, and 26890625 is divisible by 5^6.  This is the smallest number with this property, so a(3) = 26890623.

J.-M. De Koninck, Those Fascinating Numbers, Entry 242, p. 63, Amer. Math. Soc., 2009.

Jean-Marie De Koninck, Those Fascinating Numbers, Amer. Math. Soc., (2009), page 63.

Cf. A001014, A088080, A174113.

Cf. A045882, A271443, A330480, A330481, A330486, A330483, A330484, A330485.

a(5) from Giovanni Resta, Dec 17 2019

a(6) from Giovanni Resta, Dec 19 2019

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

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A330480 Earliest start of a run of n numbers divisible by a fourth power larger than one.

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A330482 Earliest start of a run of n numbers divisible by a sixth power larger than one.

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