A068140 -id:A068140 - cp's OEIS frontend

A271446 Smallest of 6 consecutive numbers each divisible by a cube.

4379776620, 6329354875, 16620507123, 54089484125, 55072893248, 66519175371, 68769514373, 80566783622, 87372290871, 91351389622, 99156598496, 105748687372, 112806598372, 114265205871, 117243671750, 148257477247, 155970667499, 174404710246, 177398391245

Offset: 1

Giovanni Resta, Apr 26 2016

nonn

			a(1) = 4379776620 is the smallest cubeful number followed by other 5 cubeful numbers. They are divisible by 29^3, 11^3, 13^3, 3^3, 2^4, and 5^3, respectively.

Giovanni Resta, Table of n, a(n) for n = 1..2500

Cf. A046099, A068140, A122692, A271444, A271445, A271447, A271443.

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

Original entry on oeis.org

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.

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

A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 567, 624, 728, 783, 944, 1160, 1431, 1592, 1624, 1647, 1808, 1863, 2024, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3087, 3320, 3429, 3536, 3591, 3624, 3752, 3992, 4023, 4184, 4239, 4375, 4400, 4455, 4624, 4671, 4887, 4912, 4913, 5048, 5103

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A072911(k) > 1, all the terms of A340152 would be in this sequence.

In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 4912, 5750, 6858, ...

			80 is a term since A072911(80) = A072911(81) = 2.

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

Cf. A072911, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349308.

f[p_, e_] := EulerPhi[e]; ephi[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[5000], ephi[#] == ephi[# + 1] > 1 &]

A342187 Numbers k such that both k and k+1 are not exponentially odd numbers.

Original entry on oeis.org

44, 48, 49, 63, 75, 80, 98, 99, 116, 147, 171, 175, 207, 244, 260, 275, 288, 315, 324, 332, 360, 363, 368, 387, 404, 475, 476, 495, 507, 524, 528, 531, 539, 548, 549, 575, 603, 604, 624, 636, 656, 675, 692, 724, 725, 747, 764, 774, 800, 819, 832, 844, 845, 846

Offset: 1

Amiram Eldar, Mar 04 2021

nonn

The numbers of terms not exceeding 10^k for k = 2, 3, ..., are 8, 64, 624, 6281, 62779, 627904, 6279725, 62796307, 627961560, ... Apparently this sequence has an asymptotic density 0.062796...

			44 is a term since 44 = 2^2 * 11 and 45 = 3^2 * 5 both have an even exponent in their prime factorization.

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

Cf. A072587, A268335.

Similar sequences: A068140, A068781, A342188, A342189.

expOddQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], OddQ]; Select[Range[10^3], !expOddQ[#] && !expOddQ[# + 1] &]

A342188 Numbers k such that both k and k+1 are not exponentially squarefree numbers.

Original entry on oeis.org

80, 624, 2511, 5264, 6399, 7695, 7856, 10287, 13040, 14640, 15471, 15632, 18063, 19375, 20624, 20816, 23247, 23408, 25839, 27135, 28560, 28592, 31023, 31184, 33615, 35072, 36015, 36368, 38799, 38960, 39375, 40816, 41391, 44144, 46250, 46575, 46736, 49167, 51920

Offset: 1

Amiram Eldar, Mar 04 2021

nonn

The numbers of terms not exceeding 10^k for k = 2, 3, ..., are 1, 2, 7, 72, 719, 7226, 72238, 722565, 7225651, ... Apparently this sequence has an asymptotic density 0.00007225...

			80 is a term since 80 = 2^4 * 5 and 81 = 3^4 both have a nonsquarefree exponent in their prime factorization.

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

Cf. A130897, A209061.

Similar sequences: A068140, A068781, A342187, A342189.

expSqFQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], SquareFreeQ]; Select[Range[5*10^4], !expSqFQ[#] && !expSqFQ[# + 1] &]

A342189 Numbers k such that both k and k+1 are not exponentially 2^n-numbers.

Original entry on oeis.org

135, 296, 343, 351, 375, 512, 728, 999, 1160, 1215, 1375, 1431, 1592, 1624, 2079, 2240, 2295, 2375, 2456, 2624, 2727, 2888, 2943, 3104, 3159, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4616, 4671, 4832, 4887, 4913, 5048, 5144, 5319, 5480, 5535, 5696, 5831, 6183

Offset: 1

Amiram Eldar, Mar 04 2021

nonn

The numbers of terms not exceeding 10^k for k = 3, 4, ..., are 8, 76, 775, 7776, 77845, 778303, 7783285, 77832769, ... Apparently this sequence has an asymptotic density 0.0077832...

			135 is a term since 135 = 3^3 * 5 and 136 = 2^3 * 17 both have an exponent in their prime factorization which is not a power of 2.

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

cf. A138302.

Similar sequences: A068140, A068781, A342187, A342188.

exp2nQ[n_] := AllTrue[FactorInteger[n][[;;, 2]], # == 2^IntegerExponent[#, 2] &]; Select[Range[10^4], ! exp2nQ[#] && ! exp2nQ[# + 1] &]

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

Original entry on oeis.org

80, 135, 296, 343, 375, 624, 728, 1160, 1431, 1592, 1624, 2240, 2295, 2456, 2511, 2624, 2727, 2888, 3429, 3591, 3624, 3752, 3992, 4023, 4184, 4671, 4887, 4913, 5048, 5144, 5264, 5319, 5480, 5696, 6183, 6344, 6375, 6591, 6615, 6776, 6858, 6859, 7479, 7624, 7640

Offset: 1

Amiram Eldar, Nov 14 2021

nonn

Without the restriction that A321167(k) > 1, all the terms of A340152 would be in this sequence.

In contrast to A001274, which has only one known pair of consecutive terms (5186 and 5187), this sequence seems to have many pairs of consecutive terms. The smaller members of these pairs are 6858, 13375, 22625, ...

			80 is a term since A321167(80) = A321167(81) = 3.

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

Cf. A321167, A340152.
Subsequence of A068140.
Similar sequences: A001274, A287055, A293184, A326403, A349307.

f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; fe[p_, e_] := uphi[e]; euphi[n_] := Times @@ fe @@@ FactorInteger[n]; Select[Range[8000], euphi[#] == euphi[# + 1] > 1 &]

A365867 Numbers k such that k and k+1 are both divisible by the cube of their least prime factor.

Original entry on oeis.org

80, 135, 296, 343, 351, 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, 2672, 2727, 2888, 2943, 3104, 3159, 3320, 3375, 3536, 3591, 3624, 3752, 3807, 3968, 4023, 4184

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that k and k+1 are both terms of A365866.

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are , 12, 110, 1119, 11167, 111662, 1116693, 11166978, 111669826, 1116697990, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0111669... .

			80 is a term since 2 is the least prime factor of 80 and 80 is divisible by 2^3 = 8, and 3 is the least prime factor of 81 and 81 is divisible by 3^3 = 27.

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

Cf. A067029.
Subsequence of A068140 and A365866.
A365868 is a subsequence.

q[n_] := FactorInteger[n][[1, -1]] >= 3; consec[kmax_] := Module[{m = 1, c = Table[False, {2}], s = {}}, Do[c = Join[Rest[c], {q[k]}]; If[And @@ c, AppendTo[s, k - 1]], {k, 1, kmax}]; s]; consec[5000]

lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = factor(k)[1,2] >= 3; if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A365868 Starts of runs of 3 consecutive integers that are divisible by the cube of their least prime factor.

Original entry on oeis.org

1375, 16119, 25623, 28375, 52623, 55375, 57967, 79623, 82375, 90207, 94471, 106623, 109375, 129623, 133623, 134431, 136375, 160623, 163375, 164295, 187623, 190375, 206143, 214623, 217375, 241623, 244375, 268623, 271375, 280231, 295623, 298375, 312471, 322623, 325375

Offset: 1

Amiram Eldar, Sep 21 2023

Numbers k such that k, k+1 and k+2 are all terms of A365866.
Numbers of the form 4*k+2 are not terms of A365866. Therefore, there are no runs of 4 or more consecutive integers. Since the middle integer in each triple is divisible by 8, all the terms of this sequence are of the form 8*k+7.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are , 1, 11, 109, 1092, 10899, 109125, 1091335, 10912897, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0001091... .

			1375 is a term since 5 is the least prime factor of 1375 and 1375 is divisible by 5^3 = 125, 2 is the least prime factor of 1376 and 1376 is divisible by 2^3 = 8, and 3 is the least prime factor of 1377 and 1377 is divisible by 3^3 = 27.

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

Cf. A067029.

Subsequence of A004771, A068140, A365866 and A365867.

Select[8 * Range[41000] + 7, AllTrue[# + {0, 1, 2}, FactorInteger[#1][[1, -1]] >= 3 &] &]

is(n) = factor(n)[1,2] >= 3;
lista(kmax) = forstep(k = 7, kmax, 8, if(is(k) && is(k+1) && is(k+2), print1(k, ", ")));

cp's OEIS Frontend

A271446 Smallest of 6 consecutive numbers each divisible by a cube.

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

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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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A349307 Numbers k such that A072911(k) = A072911(k+1) > 1.

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Mathematica

A342187 Numbers k such that both k and k+1 are not exponentially odd numbers.

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Mathematica

A342188 Numbers k such that both k and k+1 are not exponentially squarefree numbers.

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Mathematica

A342189 Numbers k such that both k and k+1 are not exponentially 2^n-numbers.

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Mathematica

A349308 Numbers k such that A321167(k) = A321167(k+1) > 1.

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Mathematica

A365867 Numbers k such that k and k+1 are both divisible by the cube of their least prime factor.