A122692 -id:A122692 - 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

A271443 Earliest start of a run of n numbers divisible by a cube larger than one.

Original entry on oeis.org

8, 80, 1375, 22624, 18035622, 4379776620, 1204244328624, 2604639091138248, 2604639091138248

Offset: 1

Giovanni Resta, Apr 23 2016

a(5)-a(7) were found by Donovan Johnson.

			a(9) = 2604639091138248 and the following 8 numbers are divisible by 2^3, 11^3, 5^3, 17^3, 7^3, 13^3, 3^3, 19^3, and 2^4, respectively.

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

Cf. A046099, A068140, A122692, A271444, A271445, A271446, A271447.

Cf. A045882, A330480, A330481, A330482, A330486, A330483, A330484, A330485.

a[n_] := Block[{k=1, c=0}, While[ c

A271444 Smallest of 4 consecutive numbers each divisible by a cube greater than one.

Original entry on oeis.org

22624, 355374, 885624, 912247, 1558248, 1642624, 1728375, 1761991, 2068373, 2485375, 2948373, 2987872, 3072248, 3073623, 3243750, 3571749, 3744872, 3772248, 3916374, 4231248, 4442877, 4503247, 4730373, 4757750, 5301125, 5344623, 5516125, 5812477, 6017247

Offset: 1

Giovanni Resta, Apr 26 2016

nonn

			a(1)=22624 is the smallest cubeful number followed by other 3 cubeful numbers. They are divisible by 2^5, 5^3, 3^3, and 11^3, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A046099, A068140, A122692, A271445, A271446, A271447, A271443.

cubQ[n_] := Max[Last /@ FactorInteger[n]] > 2; Select[Range[10^6], cubQ[#] && cubQ[# + 1] && cubQ[# + 2] && cubQ[# + 3] &]
SequencePosition[Table[If[AnyTrue[Rest[Divisors[n]],IntegerQ[CubeRoot[#]]&],1,0],{n,61*10^5}],{1,1,1,1}][[;;,1]] (* Harvey P. Dale, Jan 05 2025 *)

Definition clarified by Harvey P. Dale, Jan 05 2025

A271445 Smallest of 5 consecutive numbers each divisible by a cube.

Original entry on oeis.org

18035622, 100942496, 133799496, 146447622, 156406624, 185966872, 192779375, 215927748, 314066750, 327879871, 363664375, 377956500, 403254124, 412284624, 422615124, 440799246, 458147500, 520659248, 558732248, 562037373, 634965372, 642252750, 664596248

Offset: 1

Giovanni Resta, Apr 26 2016

nonn

			a(1) = 18035622 is the smallest cubeful number followed by other 4 cubeful numbers. They are divisible by 3^4, 17^3, 2^3, 5^4, and 7^3, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A046099, A068140, A122692, A271444, A271446, A271447, A271443.

A271446 Smallest of 6 consecutive numbers each divisible by a cube.

Original entry on oeis.org

4379776620, 6329354875, 16620507123, 54089484125, 55072893248, 66519175371, 68769514373, 80566783622, 87372290871, 91351389622, 99156598496, 105748687372, 112806598372, 114265205871, 117243671750, 148257477247, 155970667499, 174404710246, 177398391245

Offset: 1

Giovanni Resta, Apr 26 2016

nonn

			a(1) = 4379776620 is the smallest cubeful number followed by other 5 cubeful numbers. They are divisible by 29^3, 11^3, 13^3, 3^3, 2^4, and 5^3, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..2500

Cf. A046099, A068140, A122692, A271444, A271445, A271447, A271443.

A271447 Smallest of 7 consecutive numbers each divisible by a cube.

Original entry on oeis.org

1204244328624, 4224987665871, 17911333617875, 18105599700248, 20656510708125, 20917131156124, 21707874550623, 30199064929748, 30517770625623, 32526295907749, 43865182834744, 47130022943124, 48617303189245, 50499660546373, 53555917697500, 53971309892123

Offset: 1

Giovanni Resta, Apr 26 2016

nonn

			a(1) = 1204244328624 is the smallest cubeful number followed by other 4 cubeful numbers. They are divisible by 2^4, 5^3, 19^3, 3^3, 11^4, 37^3, and 7^3, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..600

Cf. A046099, A068140, A122692, A271444, A271445, A271446, A271443.

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

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A271443 Earliest start of a run of n numbers divisible by a cube larger than one.

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A271444 Smallest of 4 consecutive numbers each divisible by a cube greater than one.

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A271445 Smallest of 5 consecutive numbers each divisible by a cube.

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A271446 Smallest of 6 consecutive numbers each divisible by a cube.

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A271447 Smallest of 7 consecutive numbers each divisible by a cube.

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