A068785 -id:A068785 - 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

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 &]

A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

Original entry on oeis.org

9, 569, 49689, 4937225, 493490641, 49348095737, 4934805110729, 493480252693889, 49348022079085897, 4934802199975704129, 493480220066583590433, 49348022005552308828457, 4934802200546833521392241, 493480220054489318828539601, 49348022005446802425711456713, 4934802200544679211736756034457

Offset: 0

Seiichi Manyama, Jun 21 2024

nonn

Wikipedia, Jacobi four square theorem

Cf. A068785, A373881, A373883, A373884, A373885.

Cf. A024916, A046895.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(4, 10^n);

from math import isqrt
def A373882(n): return 1+((-(s:=isqrt(a:=10**n))**2*(s+1)+sum((q:=a//k)*((k<<1)+q+1) for k in range(1,s+1))&-1)<<2)+(((t:=isqrt(m:=a>>2))**2*(t+1)-sum((q:=m//k)*((k<<1)+q+1) for k in range(1,t+1))&-1)<<4) # Chai Wah Wu, Jun 21 2024

a(n) = A046895(10^n).
a(n) == 1 (mod 8).
Limit_{n->oo} a(n) = Pi^2*100^n/2. - Hugo Pfoertner, Jun 21 2024

A373881 Number of lattice points inside the ball x^2 + y^2 + z^2 <= 10^n.

Original entry on oeis.org

7, 147, 4169, 132451, 4187857, 132459677, 4188781437, 132461190717, 4188790061109, 132461176423805, 4188790203273025, 132461176878317635

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373882, A373883, A373884, A373885, A373896.

Cf. A000605, A117609.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(3, 10^n);

a(n) = A117609(10^n).

Limit_{n->oo} a(n) = (4*Pi/3)*(10^n)^(3/2). - Hugo Pfoertner, Jun 21 2024

a(7)-a(10) from Hugo Pfoertner, Jun 21 2024

a(11) from Chai Wah Wu, Jun 24 2024

A373883 Number of lattice points inside or on the 5-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 10^n.

Original entry on oeis.org

11, 1903, 532509, 166711479, 52646439609, 16645828150193, 5263797438037625, 1664556518763850069, 526378909839312477785, 166455624316184206850205, 52637890147973140623040513, 16645562406807092052281075983, 5263789013922669372094091725857

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373881, A373882, A373884, A373885, A373896.

Cf. A055411, A175360.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(5, 10^n);

a(n) = A175360(10^n).

a(7) from Chai Wah Wu, Jun 22 2024
a(8)-a(10) from Chai Wah Wu, Jun 23 2024
a(11)-a(12) from Chai Wah Wu, Jun 24 2024

A373884 Number of lattice points inside or on the 6-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 10^n.

Original entry on oeis.org

13, 5757, 5260181, 5178103157, 5168770118857, 5167819662055085, 5167723229551614933, 5167713844375355566137, 5167712884142309619400885, 5167712790787647771419572729

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373881, A373882, A373883, A373885.

Cf. A175361.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(6, 10^n);

a(n) = A175361(10^n).

a(7)-a(9) from Chai Wah Wu, Jun 21 2024

A373885 Number of lattice points inside or on the 7-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 = 10^n.

Original entry on oeis.org

15, 16859, 48218513, 149732983675, 472579187185537, 1494134816265585397, 4724776275561584896709, 14941025134674593046033085

Offset: 0

Seiichi Manyama, Jun 21 2024

Cf. A068785, A373881, A373882, A373883, A373884.

Cf. A341396.

b(k, n) = my(q='q+O('q^(n+1))); polcoef((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^k/(1-q), n);
a(n) = b(7, 10^n);

a(n) = A341396(10^n).

a(7) from Chai Wah Wu, Jun 23 2024

A373896 Number of lattice points inside or on the 8-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 + x_7^2 + x_8^2 = 10^n.

Original entry on oeis.org

17, 47921, 415055025, 4068011664081, 40596481219349025, 405880555110153633585, 4058721509888208894731345, 40587130610718907618215585345, 405871222004868007901459647593809, 4058712135741827985063748936303681217

Offset: 0

Seiichi Manyama, Jun 21 2024

nonn

Cf. A068785, A373881, A373882, A373883, A373884, A373885.

Cf. A008457, A341397.

a008457(n) = sumdiv(n, d, (-1)^(n-d)*d^3);
a341397(n) = 1+16*sum(k=1, n, a008457(k));
a(n) = a341397(10^n);

a(n) = A341397(10^n).

a(n) == 1 (mod 16).

cp's OEIS Frontend

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

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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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A373882 Number of lattice points inside or on the 4-dimensional hypersphere x^2 + y^2 + z^2 + u^2 = 10^n.

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A373881 Number of lattice points inside the ball x^2 + y^2 + z^2 <= 10^n.

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A373883 Number of lattice points inside or on the 5-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 10^n.

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A373884 Number of lattice points inside or on the 6-dimensional hypersphere x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + x_6^2 = 10^n.

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