A068140 -id:A068140 - cp's OEIS frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

8, 24, 27, 44, 48, 49, 63, 75, 80, 98, 99, 116, 120, 124, 125, 135, 147, 152, 168, 171, 175, 188, 207, 224, 242, 243, 244, 260, 275, 279, 288, 296, 315, 324, 332, 342, 343, 350, 351, 360, 363, 368, 375, 387, 404, 423, 424, 440, 459, 475, 476, 495, 507, 512

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

Also numbers m such that mu(m)=mu(m+1)=0, where mu is the Moebius-function (A008683); A081221(a(n))>1. - Reinhard Zumkeller, Mar 10 2003
The sequence contains an infinite family of arithmetic progressions like {36a+8}={8,44,80,116,152,188,...} ={4(9a+2)}. {36a+9} provides 2nd nonsquarefree terms. Such AP's can be constructed to any term by solution of a system of linear Diophantine equation. - Labos Elemer, Nov 25 2002
1. 4k^2 + 4k is a member for all k; i.e., 8 times a triangular number is a member. 2. (4k+1) times an odd square - 1 is a member. 3. (4k+3) times odd square is a member. - Amarnath Murthy, Apr 24 2003
The asymptotic density of this sequence is 1 - 2/zeta(2) + Product_{p prime} (1 - 2/p^2) = 1 - 2 * A059956 + A065474 = 0.1067798952... (Matomäki et al., 2016). - Amiram Eldar, Feb 14 2021
Maximum of the n-th maximal anti-run of nonsquarefree numbers (A013929) differing by more than one. For runs instead of anti-runs we have A376164. For squarefree instead of nonsquarefree we have A007674. - Gus Wiseman, Sep 14 2024

			44 is in the sequence because 44 = 2^2 * 11 and 45 = 3^2 * 5.
From _Gus Wiseman_, Sep 14 2024: (Start)
Splitting nonsquarefree numbers into maximal anti-runs gives:
  (4,8)
  (9,12,16,18,20,24)
  (25,27)
  (28,32,36,40,44)
  (45,48)
  (49)
  (50,52,54,56,60,63)
  (64,68,72,75)
  (76,80)
  (81,84,88,90,92,96,98)
  (99)
The maxima are a(n). The corresponding pairs are (8,9), (24,25), (27,28), (44,45), etc.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kaisa Matomäki, Maksym Radziwiłł and Terence Tao, Sign patterns of the Liouville and Möbius functions, Forum of Mathematics, Sigma, Vol. 4. (2016), e14.

Cf. A068780, A068140, A068781, A068782, A068783, A068784, A068785.
Cf. A049535, A077647, A078143, A045882.
Subsequence of A261869.
Cf. A007674, A373409, A373410, A373412, A375707, A376164.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, first differences A078147.
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
Cf. A049094, A061399, A072284, A120992, A294242, A373573, A375709.

a068781 n = a068781_list !! (n-1)
a068781_list = filter ((== 0) . a261869) [1..]
-- Reinhard Zumkeller, Sep 04 2015

Select[ Range[2, 600], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 1 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 1 &]
f@n_:= Flatten@Position[Partition[SquareFreeQ/@Range@2000,n,1], Table[False,{n}]]; f@2 (* Hans Rudolf Widmer, Aug 30 2022 *)
Max/@Split[Select[Range[100], !SquareFreeQ[#]&],#1+1!=#2&]//Most (* Gus Wiseman, Sep 14 2024 *)

isok(m) = !moebius(m) && !moebius(m+1); \\ Michel Marcus, Feb 14 2021

A261869(a(n)) = 0. - Reinhard Zumkeller, Sep 04 2015

A063528 Smallest number such that it and its successor are both divisible by an n-th power larger than 1.

Original entry on oeis.org

2, 8, 80, 80, 1215, 16767, 76544, 636416, 3995648, 24151040, 36315135, 689278976, 1487503359, 1487503359, 155240824832, 785129144319, 4857090670592, 45922887663615, 157197025673216, 1375916505694208, 2280241934368767, 2280241934368767, 2280241934368767

Offset: 1

Erich Friedman, Aug 01 2001

nonn

Lesser of the smallest pair of consecutive numbers divisible by an n-th power.
To get a(j), max exponent[=A051953(n)] of a(j) and 1+a(j) should exceed (j-1).
One can find a solution for primes p and q by solving p^n*i + 1 = q^n*j; then p^n*i is a solution. This solution will be less than (p*q)^n but greater than max(p,q)^n. Thus finding the solutions for 2, 3 (p=2,q=3 and p=3,q=2), one need at most also look at 2, 5 and 3, 5. It appears that the solution with 2, 3 is always optimal. - Franklin T. Adams-Watters, May 27 2011

			a(4) = 80 since 2^4 = 16 divides 80 and 3^4 = 81 divides 81.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..100

We need A051903(a[n]) > n-1 and A051903(a[n]+1) > n-1.
Cf. A068780, A068781, A068140, A068782, A068783, A068784.
Cf. A045330, A059737.

k = 4; Do[k = k - 2; a = b = 0; While[ b = Max[ Transpose[ FactorInteger[k]] [[2]]]; a <= n || b <= n, k++; a = b]; Print[k - 1], {n, 0, 19} ]

b(n,p=2,q=3)=local(i);i=Mod(p,q^n)^-n; min(p^n*lift(i)-1,p^n*lift(-i))
a(n)=local(r);r=b(n);if(r>5^n,r=min(r,min(b(n,2,5),b(n,3,5))));r /* Franklin T. Adams-Watters, May 27 2011 */

More terms from Jud McCranie, Aug 06 2001

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

A068782 Lesser of two consecutive numbers each divisible by a fourth power.

Original entry on oeis.org

80, 624, 1215, 1376, 2400, 2511, 2672, 3807, 3968, 4374, 5103, 5264, 6399, 6560, 7695, 7856, 8991, 9152, 9375, 10287, 10448, 10624, 11583, 11744, 12879, 13040, 14175, 14336, 14640, 15471, 15632, 16767, 16928, 18063, 18224, 19359, 19375

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

The asymptotic density of this sequence is 1 - 2/zeta(4) + Product_{p prime} (1 - 2/p^4) = 0.001856185541538432217... - Amiram Eldar, Feb 16 2021

Below 9508685764, it suffices to check for n such that either n or n+1 is divisible by p^4 for some p <= 19. - Charles R Greathouse IV, Jul 17 2024

			80 is a term as 80 and 81 both are divisible by a fourth power, 2^4 and 3^4 respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A068140.

Cf. A068780, A068781, A068783, A068784, A068785, A046101, A215267.

Select[ Range[2, 25000], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 3 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 3 &]

has(n)=vecmax(factor(n)[,2])>3
is(n)=has(n+1)&&has(n) \\ Charles R Greathouse IV, Dec 19 2018

list(lim)=my(v=List(),x=1); forfactored(n=81,lim\1+1, if(vecmax(n[2][,2])>3, if(x,listput(v,n[1]-1),x=1),x=0)); Vec(v) \\ Charles R Greathouse IV, Dec 19 2018

a(0) = 0 removed by Charles R Greathouse IV, Dec 19 2018

A068783 Lesser of two consecutive numbers each divisible by a fifth power.

Original entry on oeis.org

1215, 6560, 8991, 9375, 14336, 16767, 22112, 24543, 29888, 32319, 37664, 40095, 45440, 47871, 53216, 55647, 60992, 63423, 68768, 71199, 76544, 78975, 84320, 86751, 90624, 92096, 94527, 99872, 102303, 107648, 109375, 110079, 115424

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

The asymptotic density of this sequence is 1 - 2/zeta(5) + Product_{p prime} (1 - 2/p^5) = 0.000284512101137896862... - Amiram Eldar, Feb 16 2021

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

Cf. A068780, A068781, A068140, A068782, A068784, A068785.

Select[ Range[2, 250000], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 4 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 4 &]
SequencePosition[Table[If[Max[FactorInteger[n][[All,2]]]>4,1,0],{n,120000}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 25 2018 *)

A068784 Lesser of two consecutive numbers each divisible by a sixth power.

Original entry on oeis.org

16767, 29888, 63423, 76544, 109375, 110079, 123200, 156735, 169856, 203391, 216512, 250047, 263168, 296703, 309824, 343359, 356480, 390015, 403136, 436671, 449792, 483327, 496448, 529983, 543104, 576639, 589760, 623295, 636416, 669951

Offset: 1

Robert G. Wilson v, Mar 04 2002

nonn

The asymptotic density of this sequence is 1 - 2/zeta(6) + Product_{p prime} (1 - 2/p^6) = 0.000045351901298014669... - Amiram Eldar, Feb 16 2021

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

Cf. A068780, A068781, A068140, A068782, A068783, A068785.

Select[ Range[2, 10^6], Max[ Transpose[ FactorInteger[ # ]] [[2]]] > 5 && Max[ Transpose[ FactorInteger[ # + 1]] [[2]]] > 5 &]

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

A176313 First of two consecutive numbers with at least one 3 in their prime signature.

Original entry on oeis.org

135, 296, 343, 375, 999, 1160, 1431, 1592, 1624, 2295, 2375, 2456, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4887, 4913, 5048, 5144, 5319, 5480, 5831, 6183, 6344, 6375, 6615, 6776, 6858, 6859, 7479, 7624, 7640, 7749, 7911, 8072, 8375, 8775, 8936, 9125, 9207, 9368, 9624, 10071, 10232, 10375, 10503, 10632, 10664, 10984, 11124, 11319, 11367, 11528, 11624, 11799, 11960

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 07 2010

nonn

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17.

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

A068140 lists the smallest of two consecutive numbers such that each is divisible by a cube greater than 1. See also A000578, A001235, A176297, A176350.

f[n_]:=MemberQ[Last/@FactorInteger[n], 3]; Select[Range[8!],f[#]&&f[#+1]&]

Edited by Matthew Vandermast, Dec 09 2010

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.

cp's OEIS Frontend

A068781 Lesser of two consecutive numbers each divisible by a square.

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A063528 Smallest number such that it and its successor are both divisible by an n-th power larger than 1.

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

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A068782 Lesser of two consecutive numbers each divisible by a fourth power.

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A068783 Lesser of two consecutive numbers each divisible by a fifth power.

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A068784 Lesser of two consecutive numbers each divisible by a sixth power.

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A122692 Cubeful numbers whose neighbors are also cubeful.

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