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 1-6 of 6 results.

A067813 Start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 3.

Original entry on oeis.org

8, 27, 170, 602, 2522, 211673
Offset: 1

Views

Author

Shyam Sunder Gupta, Feb 07 2002

Keywords

Comments

602 is the first number having 4 and 5 consecutive integers with 3 prime factors. - T. D. Noe, Mar 19 2014

Examples

			a(4)=602 because 602 is the start of a record breaking run of 5 consecutive integers (602 to 606) each having 3 prime factors; i.e. bigomega(n)=A001222(n)=3 for n = 602, ..., 606.

Crossrefs

Cf. A007304, A045984, A067814, A067820, A067821, A067822, A117969, A259756.

Programs

  • Mathematica
    bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==3, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]
Module[{nn=8,po},po=PrimeOmega[Range[5000000]];Flatten[Table[ SequencePosition[ po,PadRight[{},n,3],1],{n,nn}],1]][[All,1]]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2019 *)
  • PARI
    show(lim)=my(was,r,ct); forfactored(n=2, lim\1+1, is=vecsum(n[2][, 2])==3; if(is, ct++; if(ct>r, r=ct; print(r" "n[1]-r+1)),ct=0)) \\ Charles R Greathouse IV, Jun 26 2019

Extensions

Edited by Dean Hickerson, Jul 31 2002

A067820 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 5.

Original entry on oeis.org

32, 944, 15470, 57967, 632148, 14845324, 69921004, 888781058, 2674685524, 10077383364, 21117216104, 393370860205, 3157222675953, 5509463413255, 24819420480104, 361385490681003, 441826936079342
Offset: 1

Views

Author

Shyam Sunder Gupta, Feb 07 2002

Keywords

Comments

a(16) > 3*10^13. - Brian Trial, May 13 2017
All multiples of 32 greater than 32 are of form 2^5*m and have at least 6 factors. Thus this sequence will be limited to a run of at most 31 integers. - Brian Trial, May 13 2017
a(18) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

Examples

			a(3)=15470 because 15470 is the start of a record breaking run of 3 consecutive integers (15470 to 15472) each having 5 prime factors; i.e. bigomega(n)=A001222(n)=5 for n = 15470, ..., 15472.

Crossrefs

Cf. A067813, A067814, A067821, A067822.
Subsequence of A014614.

Programs

  • Mathematica
    bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==5, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]
Table[SequencePosition[PrimeOmega[Range[15*10^6]],PadRight[{},n,5],1][[All,1]],{n,6}]//Flatten (* The program generates the first six terms of the sequence. *) (* Harvey P. Dale, Sep 03 2022 *)

Extensions

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(13)-a(14) from Donovan Johnson, Jan 31 2009
a(15) from Brian Trial, May 13 2017
a(16)-a(17) from Toshitaka Suzuki, Aug 31 2025

A067814 The start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 4.

Original entry on oeis.org

16, 135, 1274, 4023, 12122, 204323, 355923, 3405122, 49799889, 202536181, 3195380868, 5208143601, 85843948321, 97524222465
Offset: 1

Views

Author

Shyam Sunder Gupta, Feb 07 2002

Keywords

Comments

3405122 is the first number having 8 and 9 consecutive integers with 4 prime factors. - T. D. Noe, Mar 19 2014

Examples

			a(4)=4023 because 4023 is the start of a record breaking run of 4 consecutive integers (4023 to 4026) each having 4 prime factors; i.e. bigomega(n)=A001222(n)=4 for n = 4023, ..., 4026.

Crossrefs

Cf. A067813, A067820, A067821, A067822.

Programs

  • Mathematica
    bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==4, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]

Extensions

Edited by Dean Hickerson, Jul 31 2002
More terms from Don Reble, Aug 11 2002, who remarks that the sequence is now complete.

A067822 The start of a record-breaking run of consecutive integers with a number of prime factors equal to 7.

Original entry on oeis.org

128, 29888, 3145310, 296299374, 15605704374, 242576758750, 1981162639374, 126460514648223
Offset: 1

Views

Author

Shyam Sunder Gupta, Feb 07 2002

Keywords

Examples

			a(3) = 3145310 because 3145310 is the start of a record breaking run of 3 consecutive integers (3145310 to 3145312) each having 7 prime factors; i.e., bigomega(n) = A001222(n) = 7 for n = 3145310, ..., 3145312.

Crossrefs

Cf. A001222, A067813, A067814, A067820, A067821.

Programs

  • Mathematica
    bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==7, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]

Extensions

Edited by Dean Hickerson, Jul 31 2002
More terms from Jens Kruse Andersen, Aug 23 2003
a(7) from Donovan Johnson, Jan 31 2009
a(8) from Brian Trial, Jun 28 2017

A117969 Start of least run of maximal length of consecutive n-almost primes.

Original entry on oeis.org

2, 33, 211673, 97524222465
Offset: 1

Views

Author

Rick L. Shepherd, Apr 05 2006

Keywords

Comments

For n>=2 there cannot be more than 2^n - 1 consecutive n-almost primes. Is it known whether there always exists such a run of length 2^n - 1? If not, I conjecture so. This is confirmed to be true for terms through a(4). Terms here equal the last terms of corresponding finite sequences: a(3) = A067813(6). a(4) was computed by Don Reble as A067814(14). a(5) >= A067820(12).
a(4) is smaller than the number 488995430567765317569 found by Forbes. [From T. D. Noe, Oct 29 2008]

Examples

			a(2) = 33 because 33, 34, 35 is the least run of three consecutive 2-almost primes (semiprimes).

Links

Crossrefs

Cf. A067813, A067814, A067820, A067821, A067822.

A337463 Numbers k such that k, k+1, k+2 and k+3 are products of 6 primes (A046306).

Original entry on oeis.org

8706123, 24463374, 32442848, 32942943, 36782289, 48580623, 55486248, 57476573, 59600365, 59757774, 62481222, 62664810, 62884590, 63262374, 63728124, 64724373, 65159575, 65450824, 69362487, 70302087, 70370223, 70785924, 71494773, 72060272, 72503682, 73256910, 73638422, 74066874, 74361858
Offset: 1

Views

Author

Zak Seidov, Sep 29 2020

Keywords

Comments

a(1) = 8706123 = A067821(4).

Examples

			8706123 = 3^4*19*5657, 8706124 = 2*2*7*7*43*1033,
8706125 = 5^3*17*17*241, 8706126 = 2*3*11*13*73*139.

Crossrefs

Cf. A067821, A046306, A237201.

Programs

  • PARI
    for(k=2^6,75000000,my(found=1);for(j=0,3,if(bigomega(k+j)!=6,found=0;break));if(found,print1(k,", "))) \\ Hugo Pfoertner, Oct 21 2020
Showing 1-6 of 6 results.

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