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 1-5 of 5 results.

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

Original entry on oeis.org

254540, 310155, 378014, 421134, 432795, 483405, 486590, 486794, 488565, 489345, 507129, 522444, 545258, 549185, 558789, 558830, 567644, 577940, 584154, 591260, 598689, 627095, 634809, 637329, 663585, 666995, 667029, 678755, 687939, 690234
Offset: 1

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Author

Artur Jasinski, May 07 2008

Keywords

Comments

For the smallest number r such that r and r+1 have n distinct prime factors, see A093548.
Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016
Subsequence of the variant A321505 defined with "at least 5" instead of "exactly 5" distinct prime factors. See A321495 for the differences. - M. F. Hasler, Nov 12 2018
The subset of numbers where n and n+1 are also squarefree gives A318964. - R. J. Mathar, Jul 15 2023

Crossrefs

Programs

  • Mathematica
    a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
    Transpose[SequencePosition[Table[If[PrimeNu[n]==5,1,0],{n,700000}],{1,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2015 *)
  • PARI
    is(n)=omega(n)==5 && omega(n+1)==5 \\ Charles R Greathouse IV, Jun 02 2016

Formula

{k: k in A051270 and k+1 in A051270}. - R. J. Mathar, Jul 19 2023

A321506 Numbers m such that m and m+1 each have at least 6 distinct prime factors.

Original entry on oeis.org

11243154, 13516580, 16473170, 16701684, 17348330, 19286805, 20333495, 21271964, 21849905, 22054515, 22527141, 22754589, 22875489, 24031370, 25348070, 25774329, 28098245, 28618394, 28625960, 30259229, 31846269, 32642805, 32734910, 33205029, 33631520, 33641894, 35023365
Offset: 1

Views

Author

M. F. Hasler, Nov 12 2018

Keywords

Comments

Equals A273879 up to a(138) = 58524465, which is not in A273879: see A321496 for the complement.

Crossrefs

Cf. A273879 (variant with "exactly 6"), A321496 (terms not in A273879).
Cf. A321505 (analog for k=5 prime factors).

Programs

A321504 Numbers k such that k and k+1 each have at least 4 distinct prime factors.

Original entry on oeis.org

7314, 8294, 8645, 9009, 10659, 11570, 11780, 11934, 13299, 13629, 13845, 14420, 15105, 15554, 16554, 16835, 17204, 17390, 17654, 17765, 18095, 18290, 18444, 18920, 19005, 19019, 19095, 19227, 20349, 20405, 20769, 21164, 21489, 21735, 22010, 22154, 22659, 23001, 23114, 23484, 23529, 23540, 23919, 24395
Offset: 1

Views

Author

M. F. Hasler, Nov 12 2018

Keywords

Comments

Equals A140078 up to a(123) but a({124, 214, 219, 276, 321, 415, ...}) = { 38570, 51414, 51765, 58695, 62985, 71070, ...} are not in A140078, see A321494.

Crossrefs

Cf. A321505, A321506 (variant for k=5 & k=6 prime factors).

Programs

  • Mathematica
    SequencePosition[Table[If[PrimeNu[n]>3,1,0],{n,25000}],{1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 29 2019 *)
  • PARI
    is(n)=omega(n)>=4&&omega(n+1)>=4

A321495 Numbers k such that k and k+1 have at least 5 but not both exactly 5 distinct prime factors.

Original entry on oeis.org

728364, 1565564, 1774409, 1817529, 1923635, 2162094, 2187185, 2199834, 2225894, 2369850, 2557190, 2594514, 2659734, 2671305, 2794154, 2944689, 2964884, 3126045, 3139730, 3170244, 3244955, 3273809, 3279639, 3382379, 3387054, 3506810, 3555110, 3585945, 3686969, 3711630
Offset: 1

Views

Author

M. F. Hasler, Nov 12 2018

Keywords

Comments

Complement of A140079 (k and k+1 have exactly 5 distinct prime factors) in A321505 (k and k+1 have at least 5 distinct prime factors).

Crossrefs

Cf. A140079, A321505; A321494, A321496 (analog for 4 & 6 factors).

Programs

  • Mathematica
    aQ[n_]:=Module[{v={PrimeNu[n],PrimeNu[n+1]}},Min[v]>4 && v!={5,5}]; Select[Range[120000], aQ] (* Amiram Eldar, Nov 12 2018 *)
  • PARI
    is(n)=vecmin(n=[omega(n), omega(n+1)])>4&&n!=[5,5]

Formula

A321497 Numbers k such that both k and k+1 have at least 7 distinct prime factors and at least one has more than 7 distinct prime factors.

Original entry on oeis.org

5163068910, 5327923964, 6564937379, 6880516929, 7122669554, 8567026545, 8814635115, 9533531370, 9611079114, 10245081314, 10246336814, 10697507414, 10783550414, 10796559410, 11260076190, 11458770609, 11992960265, 12043540145, 12172828590, 12745759740, 12850545785, 12946979220
Offset: 1

Views

Author

Amiram Eldar and M. F. Hasler, Nov 13 2018

Keywords

Comments

Terms of A321489 (k and k+1 have at least 7 distinct prime factors) which don't satisfy the definition with "exactly 7".

Crossrefs

Cf. A321489, A321503, A321504, A321505, A321506, A321493, A321494, A321495, A321496 (analog for 3 .. 6 factors).

Programs

  • Mathematica
    aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>6 && v!={7, 7}]; Select[Range[10^10], aQ]
  • PARI
    is(n)=omega(n)>6&&omega(n+1)>6&&(omega(n)>7||omega(n+1)>7)
Showing 1-5 of 5 results.