A045882 -id:A045882 - cp's OEIS frontend

A330484 Earliest start of a run of n numbers divisible by a ninth power larger than one.

512, 3995648, 2889212890624, 18705093636361328125, 19810215665260426138787109374

Offset: 1

Jud McCranie, Dec 16 2019

a(6) <= 39109788784614310863666299138574218749. - Robert Israel, Jun 02 2020

			2889212890624 is divisible by 2^9, 2889212890625 is divisible by 5^9, and 2889212890626 is divisible by 3^9. This is the smallest number with this property, so a(3)=2889212890624.

Cf. A045882, A271443, A330480, A330481, A330482, A330486, A330483, A330485.

a(4)-a(5) from Giovanni Resta, Dec 17 2019

A330485 Earliest start of a run of n numbers divisible by a tenth power larger than one.

Original entry on oeis.org

1024, 24151040, 61938212890624, 9226967798833574218749, 13279660499584033124533574218748

Offset: 1

Jud McCranie, Dec 16 2019

			61938212890624 is divisible by 2^10, 61938212890625 is divisible by 5^10, and 61938212890626 is divisible by 3^10.  This is the smallest number with this property, so a(3)=61938212890624.

Cf. A045882, A271443, A330480, A330481, A330482, A330486, A330483, A330484.

a(4) from Giovanni Resta, Dec 17 2019

a(5) from Giovanni Resta, Dec 19 2019

A020753 Sizes of successive increasing gaps between squarefree numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19

Offset: 1

David W. Wilson

The indices of the records in A076259 are 1, 3, 6, 31, 150, 515, 13391, 131964, 664313, ... - R. J. Mathar, Jun 25 2010
Applying the test to squarefree numbers up to 10 million only produces the first ten terms of the sequence. - Harvey P. Dale, May 04 2011
Conjecture: a(n) ~ log(A020754(n))/2. - Thomas Ordowski, Jul 23 2015

			The first gap in A005117 occurs between 1 and 2 and has length 1. The next larger gap occurs between 3 and 5 and has length 2. The next larger gap is between 7 and 10 and has length 3. Etc. We are only interested in gaps that set new records.

Cf. A005117, A020754, A020755, A045882, A051681, A076259.

a := 1 ; for n from 2 do if A076259(n) > a then print(n,A076259(n)) ; a := A076259(n) ; end if; end do: # R. J. Mathar, Jun 25 2010

Union[Differences[Select[Range[10000000], SquareFreeQ]]] (* Harvey P. Dale, May 04 2011 *)

a(n) = A020755(n) - A020754(n). - M. F. Hasler, Dec 28 2015

Thanks to Christian G. Bower for additional comments.

More terms computed (using data from A020754) by M. F. Hasler, Dec 28 2015

A182174 a(n) = prime(n)^2 - n.

Original entry on oeis.org

3, 7, 22, 45, 116, 163, 282, 353, 520, 831, 950, 1357, 1668, 1835, 2194, 2793, 3464, 3703, 4470, 5021, 5308, 6219, 6866, 7897, 9384, 10175, 10582, 11421, 11852, 12739, 16098, 17129, 18736, 19287, 22166, 22765, 24612, 26531, 27850, 29889, 32000, 32719, 36438, 37205, 38764, 39555, 44474, 49681

Offset: 1

Alonso del Arte, Apr 16 2012

One way to find a run of n consecutive nonsquarefree numbers such that the first n primes appear in order as factors of numbers in the run is to use the Chinese remainder theorem (though this run is most likely not the earliest of length n).

The moduli are then of course the squares of the first n primes, while the remainders are then the first n terms of this sequence. (See A182433.)

			a(4) = 45 because the 4th prime is 7, and 7^2 - 4 = 49 - 4 = 45.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A001248 squares of primes; A045882 and A078144 pertain to runs of consecutive nonsquarefree numbers.

Cf. A014689. [Bruno Berselli, Mar 19 2013]

[NthPrime(n)^2-n: n in [1..50]]; // Bruno Berselli, Apr 16 2012

Mathematica
```
Table[Prime[n]^2 - n, {n, 50}]
```

a(n) = A000040(n)^2 - n = A001248(n) - n. - Omar E. Pol, Apr 16 2012

a(36) inserted by Vincenzo Librandi, Mar 19 2013

A107079 Minimal number of squared primes in a squarefree gap of length n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 7, 7, 7, 8, 9, 9, 10, 11, 12, 12, 13, 13, 14, 14, 15, 16, 17, 17, 17, 18, 18, 18, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 30, 31, 32, 32, 32, 32, 33, 33, 34, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 40, 41, 42, 43, 43, 44, 45, 46, 46, 47

Offset: 1

Paul Barry, May 10 2005

nonn

Antti Karttunen, Table of n, a(n) for n = 1..100000
Louis Marmet, First occurrences of squarefree gaps and an algorithm for their computation
Louis Marmet, First occurrences of square-free gaps and an algorithm for their computation, arXiv preprint arXiv:1210.3829 [math.NT], 2012. - _N. J. A. Sloane_, Jan 01 2013

One more than A013928. A left inverse of A005117.

Cf. A045882, A107078.

a[n_] := Sum[Boole[SquareFreeQ[k]], {k, 1, n-1}] + 1;
Array[a, 100] (* Jean-François Alcover, Sep 11 2018, from A013928 *)

A107079(n)=1+sum(k=1,n-1,bitand(moebius(k),1)) \\ Charles R Greathouse IV, Sep 22 2008

from math import isqrt
from sympy import mobius
def A107079(n): return 1+sum(mobius(k)*((n-1)//k**2) for k in range(1,isqrt(n-1)+1)) # Chai Wah Wu, Jan 03 2024

a(n) = sum{k=0..n-1, moebius_mu(n-k-1) mod 2}.
a(n) = A013928(n+1) + A107078(n).
From Antti Karttunen, Oct 07 2016: (Start)
a(n) = 1 + A013928(n). [Cf. Charles R Greathouse IV's PARI-program.]
For all n >= 1, a(A005117(n)) = n. (End)

New definition from Charles R Greathouse IV, Sep 22 2008

A069021 Start of the first occurrence of n consecutive numbers divisible by a square greater than 1.

Original entry on oeis.org

1, 8, 48, 242, 844, 22020, 217070, 1092747, 8870024, 221167422, 221167422, 47255689915, 82462576220, 1043460553364, 79180770078548, 3215226335143218, 23742453640900972

Offset: 1

Amarnath Murthy, Apr 02 2002

nonn

Apart from first term, identical to A045882.

a = {1, 1}; k = 4; Do[ While[ Min[a] < 2, k++; a = Drop[a, 1]; a = Append[ a, Max[ Transpose[ FactorInteger[k]] [[2]] ]]]; Print[k - n + 1]; k++; a = Append[ a, Max[ Transpose[ FactorInteger[k]] [[2]] ]], {n, 2, 9}]

Definition clarified by Harvey P. Dale, Aug 21 2022

A088080 Start of n successive numbers divisible by n-th powers.

Original entry on oeis.org

1, 8, 1375, 202099373, 105636978090621, 283435321166212288109372

Offset: 1

Amarnath Murthy, Sep 22 2003

a(5) <= 105636978090621.

			a(3)=1375 because 1375=5^3*11, 1376=2^3*172, 1377=3^3*51; all multiples of cubes.

Cf. A045882, A271443, A330480, A330481, A330482, A330486, A330483, A330484, A330485.

Corrected and extended by Don Reble, Sep 27 2003
a(5) from Donovan Johnson, Dec 14 2010
a(6) from Giovanni Resta, Dec 19 2019

A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

Original entry on oeis.org

0, 4, 0, 17, 12, 16, 18, 20, 104, 0, 68, 40, 0, 89, 199, 110, 60, 127, 68, 72, 151, 161, 172, 278, 297, 0, 104, 108, 112, 849, 128, 403, 0, 579, 150, 461, 322, 164, 680, 351, 180, 561, 192, 196, 198, 819, 648, 449, 228, 232, 470, 240, 1472, 508, 521, 532, 270

Offset: 1

Gus Wiseman, Dec 09 2024

nonn

			The nonsquarefree numbers between prime(24) = 89 and prime(25) = 97 are {90, 92, 96}, so a(24) = 278.

For prime instead of nonsquarefree we have A001043.
For composite instead of nonsquarefree we have A054265.
Zeros are A068361.
A000040 lists the primes, differences A001223, seconds A036263.
A070321 gives the greatest squarefree number up to n.
A071403 counts squarefree numbers up to prime(n), restriction of A013928.
A120327 gives the least nonsquarefree number >= n.
A378086 counts nonsquarefree numbers up to prime(n), restriction of A057627.
For squarefree numbers (A005117, differences A076259) between primes:
- length is A061398, zeros A068360
- min is A112926, differences A378037
- max is A112925, differences A378038
- sum is A373197
For nonsquarefree numbers (A013929, differences A078147) between primes:
- length is A061399
- min is A377783 (differences A377784), union A378040
- max is A378032 (differences A378034), restriction of A378033 (differences A378036)
- sum is A378618 (this)
Cf. A007674, A045882, A053806, A067535, A072284, A073247, A179211, A224363, A373198, A377049, A377431.

Table[Total[Select[Range[Prime[n],Prime[n+1]],!SquareFreeQ[#]&]],{n,100}]

A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 32, 36, 40, 45, 50, 52, 54, 56, 60, 64, 68, 72, 76, 81, 84, 88, 90, 92, 96, 100, 104, 108, 112, 117, 121, 126, 128, 132, 136, 140, 144, 148, 150, 153, 156, 160, 162, 164, 169, 172, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208

Offset: 1

Gus Wiseman, Sep 15 2024

nonn

			The maximal runs of nonsquarefree numbers begin:
       4
     8   9
      12
      16
      18
      20
    24  25
    27  28
      32
      36
      40
    44  45
  48  49  50

For length instead of maximum we have A053797 (firsts A373199).
For lengths of anti-runs we have A373409 (firsts A373573).
For sum instead of maximum we have A373414, anti A373412.
For minimum instead of maximum we have A053806, anti A373410.
For anti-runs instead of runs we have A068781.
For squarefree instead of nonsquarefree we have A373415, anti A007674.
For nonprime instead of nonsquarefree we have A006093 with 2 removed.
A005117 lists the squarefree numbers, first differences A076259.
A013929 lists the nonsquarefree numbers, differences A078147, sums A329472.
A061398 counts squarefree numbers between primes, nonsquarefree A061399.
A120992 gives squarefree run-lengths, anti A373127 (firsts A373128).
A373413 adds up each maximal run of squarefree numbers, min A072284.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A000040, A020754, A045882, A049094, A054265, A077641, A101836, A143658, A294242, A375709.

Max/@Split[Select[Range[100],!SquareFreeQ[#]&],#1+1==#2&]//Most

A182433 Smallest number such that the next n integers each have the square of one of the first n primes as a factor in order.

Original entry on oeis.org

7, 547, 29347, 1308247, 652312447, 180110691547, 65335225716547, 38733853511213647, 4368761145612023947, 1804216772228848838647, 14884872991210984993091647, 9816873967836575781598117447, 143397994078495393809327283088347

Offset: 2

Alonso del Arte, Apr 28 2012

nonn

These are found by an application of the Chinese remainder theorem. The remainders are the numbers prime(n)^2 - n (A182174), and the moduli are the squares of primes (A001248).
This guarantees a run of at least n nonsquarefree numbers. But just as n! + 1 guarantees a run of at least n - 1 composite numbers, this might not be the smallest run of n nonsquarefree numbers (for that, see A045882).
Marmet credits Erick Bryce Wong with the idea of applying the Chinese remainder theorem and a sieving process to obtain upper limits for squarefree gaps. From this it occurred to me to just apply the Chinese remainder theorem to find these squarefree gaps exhibiting the squares of primes in order.
Also, beyond a(4), that is n > 4, we will observe that some of the numbers in the run of nonsquarefree numbers are divisible by more than one prime power, e.g., a(n) + 5 is divisible both by 49 (the square of the fourth prime) and 4.

			a(3) = 547 as that is the solution to the simultaneous congruences x = 3 mod 4 = 7 mod 9 = 22 mod 25. We verify that the next 3 integers meet the requirement: 548 = 4 * 137, 549 = 9 * 61, 550 = 25 * 2 * 11.
a(4) = 29347 as that is the solution to the simultaneous congruences x = 3 mod 4 = 7 mod 9 = 22 mod 25 = 45 mod 49. We verify that the next 4 integers meet the requirement: 29348 = 4 * 11 * 23 * 29, 29349 = 9 * 3 * 1087, 29350 = 25 * 2 * 587, 29351 = 49 * 599.

A. Bogomolny, Chinese Remainder Theorem from Interactive Mathematics Miscellany and Puzzles
Louis Marmet, First occurrences of squarefree gaps...

Cf. A069021, A051681.

Table[ChineseRemainder[Prime[Range[n]]^2 - Range[n], Prime[Range[n]]^2], {n, 2, 14}]

cp's OEIS Frontend

A330484 Earliest start of a run of n numbers divisible by a ninth power larger than one.

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A330485 Earliest start of a run of n numbers divisible by a tenth power larger than one.

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A020753 Sizes of successive increasing gaps between squarefree numbers.

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Maple

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A182174 a(n) = prime(n)^2 - n.

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Magma

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A107079 Minimal number of squared primes in a squarefree gap of length n.

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PARI

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A069021 Start of the first occurrence of n consecutive numbers divisible by a square greater than 1.

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A088080 Start of n successive numbers divisible by n-th powers.

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A378618 Sum of nonsquarefree numbers between prime(n) and prime(n+1).

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Mathematica

A376164 Maximum of the n-th maximal run of nonsquarefree numbers (increasing by 1 at a time).

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