A242605 -id:A242605 - cp's OEIS frontend

A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

2, 33, 1309, 27962, 3323705, 296602730, 41704979953

Offset: 1

M. F. Hasler, May 18 2014

nonn

As the example of a(4)=27962 shows, "consecutive squarefree numbers" means consecutive elements of A005117, not necessarily consecutive integers that (additionally) are squarefree; this would be a more restrictive condition.

a(8) <= 102099792179229 because A093550 - 1 is an upper bound of the present sequence.

			The two squarefree numbers following a(4)=27962, namely, 27965 and 27966, also have 4 prime divisors just as a(4).

Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

See A242605-A242608 for triples of consecutive squarefree numbers with m=2,..,5 prime factors.
See A246470 for the quadruplet and A246548 for the 5-tuple versions of this sequence.
See A039833, A066509, A176167 and A192203 for triples of consecutive numbers which are squarefree and have m=2,..,5 prime factors.

Edited and a(6)-a(7) added by Hans Havermann, Aug 27 2014

A242608 Start of a triple of consecutive squarefree numbers each of which has exactly 5 distinct prime factors.

Original entry on oeis.org

3323705, 3875934, 4393190, 4463822, 4929470, 5401626, 5654802, 6452535, 6465414, 6800934, 7427042, 7755890, 8233743, 8343906, 8406174, 8457942, 8593802, 8716323, 9186474, 9688382, 9812582, 9965415, 10364934, 10504074, 10870563, 10977834, 11460666, 11685894, 11993462, 12474602, 13151761

Offset: 1

M. F. Hasler, May 18 2014

nonn

See sequences A242605-A242607 (analog for m=2,3,4) for further information and examples; A242621 (first terms for positive m).

The definition of A192203 is more restrictive and therefore A192203 is a subsequence of this one, and A192203(1) >> A242608(1), roughly by a factor 5.

			a(1) = 3323705 = 5*7*11*89*97, a(1)+1 = 2*3*41*59*229 and a(1)+5 = 2*5*13*37*691 yield the first triple of consecutive squarefree numbers such that each of them is the product of five distinct primes.

Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

(back(n)=for(i=1,2,until(issquarefree(n--),));n);for(n=10^6,2e7,issquarefree(n)||next;ndk==ndm&&ndk==5&&omega(n)==ndm&&print1(back(n)",");ndk=ndm;ndm=omega(n))

Minor edit by Hans Havermann, Aug 19 2014

A242606 Start of a triple of consecutive squarefree numbers each of which has exactly 3 distinct prime factors.

Original entry on oeis.org

1309, 1442, 1885, 2013, 2091, 2665, 2694, 2714, 3243, 3422, 3655, 3729, 3854, 3855, 4430, 4431, 4503, 4921, 5034, 5035, 5133, 5282, 5678, 5795, 5882, 5883, 5943, 5954, 6054, 6061, 6094, 6213, 6302, 6303, 6305, 6306, 6477, 6851, 6853, 6873, 6985, 7202, 7257, 7334, 7383, 7682, 7730, 7802, 7842, 7922, 7953, 8238, 8239

Offset: 1

M. F. Hasler, May 18 2014

nonn

Sequence A066509 is a subsequence.

			The two squarefree numbers following a(1)=1309=7*11*17 are 1310=2*5*131 and 1311=3*19*23, all three have 3 prime divisors.
The same is true for a(2)=1442, 1443 and the next squarefree number which is 1446.
Further examples are provided by the first "sphenic triples" (1309, 1310, 1311), (1885, 1886, 1887) and (2013, 2014, 2015).

Harvey P. Dale, Table of n, a(n) for n = 1..10000
Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

See A242605-A242608 for triples of consecutive squarefree numbers (A005117) with m=2,...,5 prime factors; A242621 (first terms for positive m).

Transpose[Select[Partition[Select[Range[10000],SquareFreeQ],3,1], Union[ PrimeNu[ #]] == {3}&]][[1]] (* Harvey P. Dale, Apr 29 2016 *)

(back(n)=for(i=1,2,until(issquarefree(n--),));n);for(n=1,9999,issquarefree(n)||next;ndk==ndm&&omega(n)==ndm&&ndk==3&&print1(back(n)",");ndk=ndm;ndm=omega(n))

Minor edit by Hans Havermann, Aug 19 2014

A242607 Start of a triple of consecutive squarefree numbers each of which has exactly 4 distinct prime factors.

Original entry on oeis.org

27962, 37145, 39234, 42182, 50138, 51986, 58562, 62643, 64074, 83082, 84774, 89089, 95642, 120783, 123486, 133903, 134826, 146165, 149606, 153543, 159182, 166495, 170751, 176754, 177122, 178086, 178087, 179330, 180782, 203433, 207974, 211562, 212583, 214489, 219063, 219894, 219963, 225069, 228135

Offset: 1

M. F. Hasler, May 18 2014

nonn

			The two squarefree numbers following a(1)=27962, 27965 and 27966, also have 4 prime divisors just as a(1).

Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

See A242605-A242608 for squarefree triples with m = 2..5 prime factors; A242621 (first terms for positive m).

Transpose[Select[Partition[Select[Range[230000],SquareFreeQ],3,1], PrimeNu[ #] =={4,4,4}&]][[1]] (* Harvey P. Dale, Jul 06 2014 *)

(back(n)=for(i=1,2,until(issquarefree(n--),));n);for(n=1,9999,issquarefree(n)||next;ndk==ndm&&omega(n)==ndm&&ndk==4&&print1(back(n)",");ndk=ndm;ndm=omega(n))

Minor edit by Hans Havermann, Aug 19 2014

A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

Original entry on oeis.org

33, 1309, 203433, 16467033, 1990586013, 41704979953, 102099792179229

Offset: 2

Daniel Constantin Mayer, May 16 2014

The five terms for m = 2,3,4,5,6 were computed with the aid of PARI/GP. But it seems to be rather difficult to compute higher terms, if they exist at all.

The distribution of squarefree integers with exactly m prime factors is given in the book by Montgomery and Vaughan, Multiplicative Number Theory, but I do not have access to it and do not know whether it also addresses the problem of three consecutive numbers of this kind.

			33 = 3*11, 34 = 2*17, 35 = 5*7;
1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
1990586013 = 3*13*29*67*109*241, 1990586014 = 2*23*37*43*59*461, 1990586015 = 5*11*17*19*89*1259. (CPU time 2 hours and 34 minutes)

Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)

Math Overflow, Asymptotics of special square-free numbers, Mar 20 2014
Daniel Constantin Mayer, PARI/GP script "AdjacentSquareFree.gp"
Daniel Constantin Mayer, PARI/GP script "SquareFreeTriplets.gp"
Prime puzzles & problems connection, Square-free triples

Cf. A007675 (any m), A039833 (m=2), A066509 (m=3), A176167 (m=4), A192203 (m=5), A068088 (sextets with gap).

Cf. A242605-A242608 for start of triples of consecutive squarefree numbers with m=2,...,5 prime factors, A242621 for the analog of the present sequence in that spirit.

{default(primelimit,2M); lb=2; ub=2*10^9; m=1; i=0; j=0; loc=0; while(m<6, m=m+1; for(n=lb,ub, if(issquarefree(n)&&(m==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1

a(n) = A093550(n)-1. - M. F. Hasler, May 20 2014

cp's OEIS Frontend

A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A242608 Start of a triple of consecutive squarefree numbers each of which has exactly 5 distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Programs

PARI

Extensions

A242606 Start of a triple of consecutive squarefree numbers each of which has exactly 3 distinct prime factors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A242607 Start of a triple of consecutive squarefree numbers each of which has exactly 4 distinct prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A242492 For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula