A068782 -id:A068782 - 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

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

Original entry on oeis.org

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

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

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 &]

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

cp's OEIS Frontend

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

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

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

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