A051681 -id:A051681 - cp's OEIS frontend

A268313 First term of a run of at least 10 consecutive integers which are not squarefree.

221167422, 221167423, 262315467, 363504972, 463239475, 1202800371, 1407472722, 1407472723, 1557947844, 1609077723, 1911823144, 2217728772, 2695179044, 2737800168, 2847305571, 3639720042, 3639720043, 3672883247, 3865964268, 3865964269, 3982659575, 4246929267, 4818537743, 4982931368

Offset: 1

M. F. Hasler, Feb 01 2016

nonn

M. F. Hasler, Table of n, a(n) for n = 1..61

Cf. A013929, A045882 (first of the k-chains), A051681.

Cf. A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268314 (11-chains).

s10[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 9}]]; Do[If[Equal[s10[n], 0], Print[n]], {n, 10^8, 10^10}]

A268313 = { A078143[k] | A078143[k+1] = A078143[k]+1 } = { A077647[k] | A077647[k+2] = A077647[k]+2 } = { A077640[k] | A077640[k+3] = A077640[k]+3 }.

A268314 First term of a run of at least 11 consecutive integers which are not squarefree.

Original entry on oeis.org

221167422, 1407472722, 3639720042, 3865964268, 4982931368, 5005996146, 7108776620, 8044261244, 10249558974, 12766690268, 13585489166, 19792784322, 26995377572, 30410811296, 30477326444, 32070270968, 34317891368, 39956560824, 40841363528, 42216508746, 43133805944, 46295514872, 47255689915

Offset: 1

M. F. Hasler, Feb 01 2016

nonn

a(23) is the first term beginning a 12-chain. - Bill Hannaford, Oct 06 2016

Cf. A013929, A045882 (first of the k-chains), A051681.

Cf. A068781 (2-chains), A070258 (3-chains), A070284 (4-chains), A078144 (5-chains), A049535 (6-chains), A077640 (7-chains), A077647 (8-chains), A078143 (9-chains), A268313 (10-chains).

s11[x_] := Apply[Plus, Table[Abs[MoebiusMu[x+j]], {j, 0, 10}]]; Do[If[Equal[s11[n], 0], Print[n]], {n, 10^8, 10^13}]

A268314 = { A268313[k] | A268313[k+1] = A268313[k]+1 } = { A078143[k] | A078143[k+2] = A078143[k]+2 } = { A077647[k] | A077647[k+3] = A077647[k]+3 } = { A077640[k] | A077640[k+4] = A077640[k]+4 }.

a(12)-a(23) from Bill Hannaford, Oct 06 2016

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

A109473 Let m = n-th squarefree number = A005117(n), and consider the smallest pair of consecutive squarefree numbers (r,s) with gcd(r,s) = m; sequence gives values of r.

Original entry on oeis.org

1, 422, 174, 22830, 9216772051242, 234374

Offset: 1

N. J. A. Sloane, based on a suggestion from David W. Wilson, Aug 20 2005

a(7) is the first unknown value.
If m (in the table in Examples) is odd then a(m) >= A020754(m-1). If m is even then a(m) >= A020754(2m-1). - Jud McCranie, Sep 30 2020
a(12) (for m=17) is greater than 3.3*10^16. - Jud McCranie, Oct 16 2020

			n  |  m |    a(n) = r
---+----+---------------
1  |  1 |              1
2  |  2 |            422
3  |  3 |            174
4  |  5 |          22830
5  |  6 |  9216772051242
6  |  7 |         234374
7  | 10 |              ?
8  | 11 |    21971536246
9  | 13 |  8678016978774
10 | 14 |              ?
11 | 15 | 36442589727570
Specifically, 174 is squarefree, 177 is the next squarefree integer, and gcd(174,177) = 3; this is the first pair of consecutive squarefree numbers whose GCD is 3, so a(3)=174. - _Jud McCranie_, Nov 25 2019

See A109505 for another version. Cf. A005117, A051681, A020754, A337914, A337915.

a(5) from Jud McCranie, Aug 28 2005
a(8) from Jud McCranie, Aug 29 2005 (see Examples)
a(9) from Jud McCranie, Aug 31 2005 (see Examples)
Don Reble pointed out that the value of a(5), 9216772051254, given in the DATA section should have been 9216772051242, as in the EXAMPLE section. Revised definition to clarify the difference between n and m. - N. J. A. Sloane, Nov 25 2019
a(11) from Jud McCranie, Sep 30 2020 (see Examples)

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}]

A055554 An arithmetic progression each term of which is followed by at least 4 nonsquarefree consecutive integers.

Original entry on oeis.org

242, 1757711582, 3515422922, 5273134262, 7030845602, 8788556942, 10546268282, 12303979622, 14061690962, 15819402302, 17577113642, 19334824982, 21092536322, 22850247662, 24607959002, 26365670342

Offset: 1

Labos Elemer, Jul 10 2000

nonn

From the terms of A051681 or A045882 similar progressions for longer chains can be constructed.

Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A005117, A013929, A053806, A053797, A045882, A051681.

a(n) = d*n + 242, where d = LCM(242, 243, 244, 245) = 1757711340 = 2*2*3*3*3*3*3*5*7*11*11*61.

cp's OEIS Frontend

A268313 First term of a run of at least 10 consecutive integers which are not squarefree.

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A268314 First term of a run of at least 11 consecutive integers which are not squarefree.

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A020753 Sizes of successive increasing gaps between squarefree numbers.

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A109473 Let m = n-th squarefree number = A005117(n), and consider the smallest pair of consecutive squarefree numbers (r,s) with gcd(r,s) = m; sequence gives values of r.

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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.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A055554 An arithmetic progression each term of which is followed by at least 4 nonsquarefree consecutive integers.

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Author

Keywords

Comments

Links

Crossrefs

Formula