A063528 -id:A063528 - cp's OEIS frontend

A071254 Duplicate of A063528.

2, 8, 80, 80, 1215, 16767, 76544, 636416, 3995648, 24151040, 36315135, 689278976

Offset: 1

dead

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

A071318 Lesser of 2 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that both k and k+1 are in A067259.

Original entry on oeis.org

44, 49, 75, 98, 99, 116, 147, 171, 244, 260, 275, 315, 332, 363, 387, 475, 476, 507, 524, 531, 548, 549, 603, 604, 636, 692, 724, 725, 747, 764, 774, 819, 844, 845, 846, 867, 908, 924, 931, 963, 980, 1035, 1075, 1083, 1179, 1196, 1251, 1274, 1275, 1324

Offset: 1

Labos Elemer, May 29 2002

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 5, 41, 407, 4125, 41215, 412331, 4123625, 41236308, ... . Apparently, the asymptotic density of this sequence exists and equals 0.041236... . - Amiram Eldar, Jan 18 2023

The asymptotic density of this sequence is Product_{p prime} (1 - 2/p^3) - 2 * Product_{p prime} (1 - 1/p^2 - 1/p^3) + Product_{p prime} (1 - 2/p^2) = 0.041236147082334172926... . - Amiram Eldar, Jan 05 2024

			75 is a term since 75 = 3*5^2 and 76 = 2^2*19.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Subsequence of A067259, A340152.
Cf. A007674, A063528, A071619, A071620, A071621, A071622, A071623.
Cf. A008966, A051903, A212793.

a071318 n = a071318_list !! (n-1)
a071318_list = [x | x <- [1..],  a212793 x == 1, a008966 x == 0,
                    let y = x+1, a212793 y == 1, a008966 y == 0]
-- Reinhard Zumkeller, May 27 2012

With[{s = Select[Range[1350], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ Position[t, 1][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = (n>1) && (vecmax(factor(n)[, 2])==2) && (vecmax(factor(n+1)[, 2])==2); \\ Michel Marcus, Aug 02 2017

A051903(k) = A051903(k+1) = 2 when k is a term.

A059737 Lesser of the smallest pair of consecutive numbers divisible by an n-th power, but neither divisible by an (n+1)-st power.

Original entry on oeis.org

2, 44, 135, 80, 8991, 29888, 356480, 2316032, 14073344, 24151040, 326481920, 689278976, 11573190656, 76876660736, 314944159743, 2035980763136, 28996228218879, 55637069004800, 766556765683712, 1375916505694208, 19656708706009088, 129341461907898368, 2280241934368767, 787449981119234048

Offset: 0

Don Reble, May 25 2002

nonn

Cf. A063528, A045530.

More terms from Sean A. Irvine, Oct 06 2022

A174113 Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.

Original entry on oeis.org

48, 1375, 33614, 2590623, 26890623, 2372890624, 70925781248, 2889212890624, 61938212890624, 4497636425781248, 8555081787109375, 2665760081787109375, 98325140081787109375, 198816740081787109374, 11776267480163574218750, 872710687480163574218750, 50783354512519836425781248

Offset: 2

Michel Lagneau, Mar 08 2010

nonn

Least of the smallest trio of consecutive numbers divisible by an n-th power.

			a(3) = 1375 because
  1375 =  11 * 5^3;
  1376 = 172 * 2^3;
  1377 =  51 * 3^3.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1375, p. 135, Ellipses, Paris 2008.

Charles R Greathouse IV, Table of n, a(n) for n = 2..677 (next term has 1001 digits)

Cf. A068780, A068781, A068140, A068782, A068783, A068784, A045330, A059737, A063528, A051903, A051903.

with(numtheory):for n from 2 to 6 do: i:=0:for k from 1 to 3000000 while(i=0) do:j:=0:
for a from 0 to 2 do: ii:=0:for m from 1 to 4  while(ii=0) do:p:=ithprime(m)^n:if irem(k+a,p)=0 then j:=j+1:ii:=1:else fi:od:od:if j=3 then i:=1:print(k):else fi:od:od:

a(n)=my(ch,t,best=30^n);forprime(a=2, 29, forprime(b=2, 29, if(a==b,next); ch=chinese(Mod(0,a^n), Mod(-1,b^n)); if(lift(ch)>=best, next); forprime(c=2, 29, if(a==c || b==c, next); t=lift(chinese(ch, Mod(-2, c^n))); if(tCharles R Greathouse IV, Jan 16 2012

5^n < a(n) < 30^n. Can the lower bound be improved? - Charles R Greathouse IV, Jan 16 2012

a(8)-a(18) from Charles R Greathouse IV, Jan 16 2012

A071319 First of 3 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that {k, k+1, k+2} are in A067259.

Original entry on oeis.org

98, 475, 548, 603, 724, 844, 845, 1274, 1420, 1681, 1682, 1924, 2275, 2523, 2890, 3283, 3474, 3548, 3626, 3716, 4148, 4203, 4418, 4475, 4850, 4923, 4948, 5202, 5274, 5490, 5524, 5634, 5948, 6650, 6811, 6956, 7299, 7324, 7442, 7514, 7675, 8107, 8348

Offset: 1

Labos Elemer, May 29 2002

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 1, 7, 55, 570, 5628, 56174, 562151, 5621119, 56209006, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00562... . - Amiram Eldar, Jan 18 2023

The asymptotic density of this sequence is Product_{p prime} (1 - 3/p^3) - 3 * Product_{p prime} (1 - 1/p^2 - 2/p^3) + 3 * Product_{p prime} (1 - 2/p^2 - 1/p^3) - Product_{p prime} (1 - 3/p^2) = 0.0056209097169531390208... . - Amiram Eldar, Jan 12 2024

			98 is a term since 98 = 2*7^2, 99 = 3^2*11, and 100 = 2^2*5^2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Subsequence of A067259 and A071318.

Cf. A051903, A063528.

With[{s = Select[Range[10^4], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ SequencePosition[t, {1, 1}][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = (n>1) && (vecmax(factor(n)[, 2])==2) && (vecmax(factor(n+1)[, 2])==2) && (vecmax(factor(n+2)[, 2])==2); \\ Michel Marcus, Aug 02 2017

A051903(k) = A051903(k+1) = A051903(k+2) = 2 when k is a term.

A045330 Lesser of the smallest pair of consecutive numbers divisible by an n-th power, but not both divisible by an (n+1)-st power.

Original entry on oeis.org

2, 8, 135, 80, 1215, 16767, 76544, 636416, 3995648, 24151040, 36315135, 689278976, 11573190656, 1487503359, 155240824832, 785129144319, 4857090670592, 45922887663615, 157197025673216, 1375916505694208

Offset: 0

Don Reble, May 25 2002

nonn

Cf. A063528, A059737.

A071125 Least starting number initiating cubefree but nonsquarefree chain of consecutive integers with length n {j,j+1,...,j+n-1}; i.e., start of n consecutive numbers in A067259.

Original entry on oeis.org

4, 44, 98, 844, 30923, 671346, 8870025

Offset: 1

Labos Elemer, May 29 2002

Sequence is complete: multiples of 8 are not cubefree. - Donovan Johnson, Apr 27 2008

			n = 671346 = 2*3*3*13*19*151;
n = 671347 = 17*17*23*101;
n = 671348 = 2*2*47*3571;
n = 671349 = 3*7*7*4567;
n = 671350 = 2*5*5*29*463;
n = 671351 = 53*53*239.

Cf. A063528, A067259.

A051903(a(n) + j) = 2 for j = 0, 1, ..., (n-1).

a(7) from Donovan Johnson, Apr 27 2008

A071320 Least of four consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that {k, k+1, k+2, k+3} are in A067259.

Original entry on oeis.org

844, 1681, 8523, 8954, 10050, 10924, 11322, 17404, 19940, 22020, 23762, 24450, 25772, 27547, 30923, 30924, 33172, 34347, 38724, 39050, 39347, 40050, 47673, 47724, 47825, 49147, 54585, 55449, 57474, 58473, 58849, 58867, 59924, 62865

Offset: 1

Labos Elemer, May 29 2002

nonn

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 0, 0, 1, 4, 57, 555, 5492, 55078, 551443, 5512825, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000551... . - Amiram Eldar, Jan 18 2023

			k = 844 is a term since 844 = 2^2*211, k+1 = 845 = 5*13^2, k+2 = 846 = 2*3^2*47, and k+4 = 847 = 7*11^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Michael De Vlieger)

Subsequence of A067259, A071318 and A071319.

Cf. A051903, A063528.

With[{s = Select[Range[10^5], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ SequencePosition[t, {1, 1, 1}][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

A051903(k) = A051903(k+1) = A051903(k+2) = A051903(k+3) = 2 when k is a term.

A071124 Least of five consecutive numbers which are cubefree and not squarefree, i.e., {k, k+1, k+2, k+3, k+4} are in A067259.

Original entry on oeis.org

30923, 74849, 96675, 145674, 152339, 204323, 230346, 240425, 255186, 274547, 276650, 338921, 361322, 430073, 432474, 527922, 574674, 671346, 671347, 675491, 697073, 801473, 808155, 826825, 826826, 915857, 939321, 978675, 998522

Offset: 1

Labos Elemer, May 29 2002

nonn

			30923 = 17*17*107;
30924 = 2*2*3*3*859;
30925 = 5*5*1237;
30926 = 2*7*47*47;
30927 = 3*13*13*61.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A051903, A063528, A067259.

With[{s = Select[Range[10^6], And[MemberQ[#, 2], FreeQ[#, k_ /; k > 2]] &@ FactorInteger[#][[All, -1]] &]}, Function[t, Part[s, #] &@ SequencePosition[t, {1, 1, 1, 1}][[All, 1]]]@ Differences@ s] (* Michael De Vlieger, Jul 30 2017 *)

A051903(k) = A051903(k+1) = A051903(k+2) = A051903(k+3) = A051903(k+4) = 2.

cp's OEIS Frontend

A071254 Duplicate of A063528.

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

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A071318 Lesser of 2 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that both k and k+1 are in A067259.

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A059737 Lesser of the smallest pair of consecutive numbers divisible by an n-th power, but neither divisible by an (n+1)-st power.

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A174113 Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.

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A071319 First of 3 consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that {k, k+1, k+2} are in A067259.

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A045330 Lesser of the smallest pair of consecutive numbers divisible by an n-th power, but not both divisible by an (n+1)-st power.

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A071125 Least starting number initiating cubefree but nonsquarefree chain of consecutive integers with length n {j,j+1,...,j+n-1}; i.e., start of n consecutive numbers in A067259.

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A071320 Least of four consecutive numbers which are cubefree and not squarefree, i.e., numbers k such that {k, k+1, k+2, k+3} are in A067259.

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