cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

M. F. Hasler, May 18 2014

Keywords

Comments

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.

Examples

			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.

Links

Programs

  • PARI
    (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))

Extensions

Minor edit by Hans Havermann, Aug 19 2014

A242605 Start of a triple of consecutive squarefree numbers which are all semiprimes.

Original entry on oeis.org

33, 55, 85, 91, 93, 115, 118, 119, 141, 142, 143, 158, 201, 202, 203, 205, 213, 214, 215, 217, 218, 295, 298, 299, 301, 302, 323, 326, 391, 393, 411, 413, 445, 451, 511, 514, 535, 542, 551, 622, 633, 685, 694, 695, 697, 745, 763, 778, 791, 799, 815, 842, 843, 865, 898, 921, 922
Offset: 1

Views

Author

M. F. Hasler, May 18 2014

Keywords

Comments

Sequence A039833 is a subsequence.

Examples

			33 is in the sequence because 33, 34, 35 are all squarefree semiprimes.
55 is in the sequence because 55, 57, 58 (we ignore 56 because it's not squarefree) are all squarefree semiprimes.

Links

Crossrefs

Cf. A242606 (m=3), A242607 (m=4), A242608 (m=5), A242621 (first terms for positive m).

Programs

  • Mathematica
    Transpose[Select[Partition[Select[Range[1000],SquareFreeQ],3,1], Union[ PrimeOmega[ #]] =={2}&]][[1]] (* Harvey P. Dale, Feb 07 2016 *)
  • PARI
    is_A242605(n,c=2)==issquarefree(n)&&omega(n)==2&&(!c||until(issquarefree(n++),)||is_A242605(n,c-1))
  • PARI
    (back(n,c=1)=until(issquarefree(n--)&&c--,);n); for(n=1,999,issquarefree(n)||next;dk==4&&dk==dm&&numdiv(n)==dm&&print1(back(n)",");dk=dm;dm=numdiv(n))

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

Views

Author

M. F. Hasler, May 18 2014

Keywords

Comments

Sequence A066509 is a subsequence.

Examples

			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).

Links

Crossrefs

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

Programs

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

Extensions

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

Views

Author

M. F. Hasler, May 18 2014

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

Minor edit by Hans Havermann, Aug 19 2014

A246470 Start of the least quadruplet of consecutive squarefree numbers each of which has exactly n distinct prime factors.

Original entry on oeis.org

4058471, 91, 3854, 178086, 15469622, 18230787183
Offset: 1

Views

Author

Hans Havermann, Aug 27 2014

Keywords

Comments

By "consecutive squarefree numbers" we mean consecutive terms of A005117, not consecutive integers that also happen to be squarefree.

Examples

			4058471, 4058473, 4058477, and 4058479 are the smallest 4 primes that are also consecutive squarefree numbers (4058472 = 2^3*3*11*15373, 4058474 = 2*7^2*41413, 4058475 = 3*5^2*53*1021, 4058476 = 2^2*19*53401, and 4058478 = 2*3^3*17*4421), so a(1) = 4058471.
91, 93, 94, and 95 are the smallest 4 semiprimes that are also consecutive squarefree numbers, so a(2) = 91.
3854, 3855, 3857, and 3858 is the smallest 4-tuple of consecutive squarefree numbers each of which has exactly 3 prime factors, so a(3) = 3854.

Crossrefs

Cf. A005117, A242621 (triple version), A246548 (5-tuple version), A246471.

A246548 Start of the least 5-tuple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

Original entry on oeis.org

462807341, 115, 6302, 294590, 106985762
Offset: 1

Views

Author

Hans Havermann, Aug 29 2014

Keywords

Comments

By "consecutive squarefree numbers" we mean consecutive terms of A005117, not consecutive integers that also happen to be squarefree.

Examples

			462807341, 462807343, 462807347, 462807349, and 462807353 are the smallest 5 primes that are also consecutive squarefree numbers, so a(1) = 462807341.
115, 118, 119, 122, and 123 are the smallest 5 semiprimes that are also consecutive squarefree numbers, so a(2) = 115.
6302, 6303, 6305, 6306, and 6307 is the smallest 5-tuple of consecutive squarefree numbers each of which has exactly 3 prime factors, so a(3) = 6302.

Crossrefs

Cf. A005117, A242621 (triple version), A246470 (quadruplet version), A246471.

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

Views

Author

Daniel Constantin Mayer, May 16 2014

Keywords

Comments

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.

Examples

			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)

References

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

Links

Crossrefs

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.

Programs

  • PARI
    {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

Formula

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

A246471 First of the least n primes that are also consecutive terms of A005117 (squarefree numbers).

Original entry on oeis.org

2, 2, 2, 4058471, 462807341
Offset: 1

Views

Author

Hans Havermann, Aug 27 2014

Keywords

Comments

a(6) > 10^11.
a(6) <= 10211143417747. - Jens Kruse Andersen, Sep 09 2014
If the k-tuple conjecture is true then a(n) always exists. Weak upper bounds are
a(7) <= 105629504093565907896809
a(8) <= 9921941384213872059341198331469403
a(9) <= 376847848512740851019714299079899219399
a(10) <= 32108207632017215023964871615662539298039357
- Jens Kruse Andersen, Sep 10 2014

Examples

			Trivially, primes 2, 3, and 5 are consecutive squarefree numbers so a(1)-a(3) are each 2.
4058471, 4058473, 4058477, and 4058479 are the least 4 primes that are also consecutive squarefree numbers (4058472 = 2^3*3*11*15373, 4058474 = 2*7^2*41413, 4058475 = 3*5^2*53*1021, 4058476 = 2^2*19*53401, and 4058478 = 2*3^3*17*4421) so a(4) = 4058471.

Crossrefs

Cf. A005117; a(3) is first term in A242621; a(4) is first term in A246470; a(5) is first term in A246548.
Showing 1-8 of 8 results.

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