A271443 -id:A271443 - cp's OEIS frontend

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

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.

A088080 Start of n successive numbers divisible by n-th powers.

Original entry on oeis.org

1, 8, 1375, 202099373, 105636978090621, 283435321166212288109372

Offset: 1

Amarnath Murthy, Sep 22 2003

a(5) <= 105636978090621.

			a(3)=1375 because 1375=5^3*11, 1376=2^3*172, 1377=3^3*51; all multiples of cubes.

Cf. A045882, A271443, A330480, A330481, A330482, A330486, A330483, A330484, A330485.

Corrected and extended by Don Reble, Sep 27 2003
a(5) from Donovan Johnson, Dec 14 2010
a(6) from Giovanni Resta, Dec 19 2019

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).

cp's OEIS Frontend

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

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A088080 Start of n successive numbers divisible by n-th powers.

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

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