A340153 -id:A340153 - 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

A340152 Numbers k such that k and k+1 are both cubefree numbers (A004709).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 41, 42, 43, 44, 45, 46, 49, 50, 51, 52, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 82, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94

Offset: 1

Amiram Eldar, Dec 29 2020

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^3) = 0.6768927370... (A340153) (Carlitz, 1932).

			1 is a term since both 1 and 2 are cubefree numbers.
7 is not a term since 7+1 = 8 = 2^3 is not cubefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Leonard Carlitz, On a problem in additive arithmetic (II), The Quarterly Journal of Mathematics, Vol. os-3, No. 1 (1932), pp. 273-290.

Subsequence of A004709.

Subsequences: A007674, A328016.

cubefreeQ[n_] := Max @ FactorInteger[n][[;; , 2]] < 3; Select[Range[100], cubefreeQ[#] && cubefreeQ[# + 1] &]

A122692 Cubeful numbers whose neighbors are also cubeful.

Original entry on oeis.org

1376, 4375, 4913, 5751, 6859, 13311, 13376, 16120, 21249, 22625, 22626, 24353, 25624, 28376, 31375, 32751, 33615, 40473, 41743, 48249, 49625, 49735, 52624, 55376, 57968, 58375, 59751, 75249, 76625, 79624, 82376, 85375, 86751, 90208

Offset: 1

Tanya Khovanova, Oct 21 2006

nonn

The asymptotic density of this sequence is 1 - 3/zeta(3) + 3 * Product_{p prime} (1 - 2/p^3) - Product_{p prime} (1 - 3/p^3) = 1 - 3 * A088453 + 3 * A340153 - Product_{p prime} (1 - 3/p^3) = 0.00038922120241968636455... . - Amiram Eldar, Sep 12 2024

			1376 is divisible by 8, and its neighbors 1375 and 1377 are divisible by 125 and 27, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale, terms 1001..3903 from Robert Israel)

Subsequence of A046099 and A068140.

Cf. A088453, A340153.

N := 10^6: # get all terms <= N
CF := {seq(seq(x^3 * y, y = 1..floor(N/x^3)), x = 2..floor(N^(1/3)))}:
CF intersect map(`-`, CF, 1) intersect map(`+`, CF, 1): # Robert Israel, Jul 16 2014

Select[Range[2, 100000], Max[Transpose[FactorInteger[ # ]][[2]]] >= 3 && Max[Transpose[FactorInteger[# + 1]][[2]]] >= 3 && Max[Transpose[FactorInteger[# - 1]][[2]]] >= 3 &]
cnQ[{a_,b_,c_}] := And@@(# > 2 &/@{a, b, c}); Flatten[Position[Partition[Table[Max[Transpose[FactorInteger[n]][[2]]], {n, 91000}], 3, 1], ?(cnQ[#] &)]] + 1 (* _Harvey P. Dale, Jul 28 2013 *)

iscubefree(n) = vecsort(factor(n)~, 2, 4)[2, 1] < 3
s = []; for(n = 3, 200000, if(!iscubefree(n - 1) && !iscubefree(n) && !iscubefree(n + 1), s = concat(s, n))); s \\ Colin Barker, Jul 16 2014

A051903(n)=if(n>1, vecmax(factor(n)[, 2]), 0)
is(n)=A051903(n)>2 && A051903(n-1)>2 && A051903(n+1)>2 \\ Charles R Greathouse IV, Jul 23 2014

A349236 Gaps between cubefree numbers: a(n) = A004709(n+1) - A004709(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

This sequence is unbounded since by the Chinese Remainder Theorem there are arbitrarily long runs of consecutive numbers that are not cubefree.
The first occurrence of a(n) = 1, 2, ... is at n = 1, 7, 68, 1145, 18825, 15003967, ...
The asymptotic density of the occurrences of 1 in this sequence is density(A340152)/density(A004709) =  A340153/A088453 = 0.8136635872...

			a(1) = A004709(2) - A004709(1) = 2 - 1 = 1.
a(7) = A004709(8) - A004709(7) = 9 - 7 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Cf. A002117, A046099, A004709, A076259, A088453, A271443, A340152, A340153.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Differences @ Select[Range[100], cubeFreeQ]

A003557(n) = (n/factorback(factorint(n)[, 1]));
isA004709(n) = issquarefree(A003557(n));
A349236list(first_n) = { my(v=vector(first_n),k=0,e=1); for(n=2,oo,if(isA004709(n),k++; v[k] = n-e; e = n); if(#v==k, return(v))); }; \\ Antti Karttunen, Nov 11 2021

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3) (A002117).

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

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A340152 Numbers k such that k and k+1 are both cubefree numbers (A004709).

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A122692 Cubeful numbers whose neighbors are also cubeful.

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A349236 Gaps between cubefree numbers: a(n) = A004709(n+1) - A004709(n).

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