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.

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A046412 Lengths of nonsquarefree repunits.

Original entry on oeis.org

9, 18, 22, 27, 36, 42, 44, 45, 54, 63, 66, 72, 78, 81, 84, 88, 90, 99, 108, 110, 111, 117, 126, 132, 135, 144, 153, 154, 156, 162, 168, 171, 176, 180, 189, 198, 205, 207, 210, 216, 220, 222, 225, 234, 242, 243, 252, 261, 264, 270, 272, 279, 286, 288, 294, 297, 306, 308, 312, 315, 324, 330, 333, 336, 342, 351, 352
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Comments

This is the set of all positive multiples of all positive members of A087094. What is the asymptotic density of this set? - Jeppe Stig Nielsen, Dec 28 2015

Links

Crossrefs

Cf. A000042, A004022, A046053, A084006, A087094.

Programs

  • Maple
    remove(t -> numtheory:-issqrfree((10^t-1)/9), [$1..90]); # Robert Israel, Dec 30 2015
  • Mathematica
    Select[Range[300],!SquareFreeQ[(10^#-1)/9]&] (* Harvey P. Dale, Jun 26 2013 *)
  • PARI
    isok(n) = ! issquarefree((10^n-1)/9); \\ Michel Marcus, Dec 31 2015

Formula

a(n)=k where (10^k-1)/9 is not squarefree. - Ray Chandler, Aug 10 2003

Extensions

Terms to a(60) from Ray Chandler, Aug 10 2003
a(61)-a(67) from Max Alekseyev, Apr 29 2022

A046415 Repunit of length a(n) has exactly 4 prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 10, 14, 41, 43, 49, 53, 109, 157, 167, 173, 197, 199, 223, 229, 269, 283, 307, 349
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Programs

  • Mathematica
    Select[Range[350],PrimeOmega[FromDigits[PadRight[{},#,1]]]==4&] (* Harvey P. Dale, Oct 27 2020 *)
  • PARI
    isok(n) = bigomega((10^n - 1)/9) == 4; \\ Michel Marcus, Apr 23 2017

Extensions

More terms from Robert Gerbicz, Nov 22 2010
Offset corrected to 1, a(18)-a(20) added by Ray Chandler, Apr 23 2017

A046416 Repunit of length a(n) has exactly 5 prime factors (counted with multiplicity).

Original entry on oeis.org

6, 25, 29, 62, 89, 134, 137, 142, 151, 179, 239, 257, 271, 277, 289
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Extensions

More terms from Robert Gerbicz, Nov 22 2010
a(14) by Bo Gyu Jeong, Jun 30 2012
a(15) from Ray Chandler, Apr 23 2017

A046417 Repunit of length a(n) has exactly 6 prime factors (counted with multiplicity).

Original entry on oeis.org

15, 16, 26, 33, 34, 39, 46, 57, 69, 76, 79, 93, 94, 106, 118, 121, 133, 169, 181, 278, 281, 293
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Extensions

More terms from Robert Gerbicz, Nov 22 2010
a(20) from Bo Gyu Jeong, Jun 30 2012
a(21)-a(22) from Ray Chandler, Apr 23 2017

A046418 Repunit of length a(n) has exactly 7 prime factors (counted with multiplicity).

Original entry on oeis.org

12, 20, 21, 22, 27, 35, 61, 65, 74, 82, 85, 141, 146, 177, 187, 194, 226, 299, 323, 329, 337
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Programs

  • Mathematica
    Select[Range[300],PrimeOmega[FromDigits[PadRight[{},#,1]]]==7&] (* Harvey P. Dale, Feb 04 2019 *)

Extensions

More terms from Robert Gerbicz, Nov 22 2010
Changed offset to 1, a(18) added by Ray Chandler, Apr 23 2017
a(19)-a(21) from Max Alekseyev, May 14 2022

A046419 Repunit of length a(n) has exactly 8 prime factors (counted with multiplicity).

Original entry on oeis.org

28, 51, 55, 58, 77, 86, 95, 98, 107, 115, 119, 124, 155, 161, 193, 209, 217, 218, 221, 233, 253, 265, 295, 298, 303, 314, 346
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Programs

  • Mathematica
    Select[Range[320],PrimeOmega[FromDigits[Table[1,#]]]==8&] (* Harvey P. Dale, Sep 04 2018 *)

Extensions

More terms from Robert Gerbicz, Nov 22 2010
Offset changed to 1, a(23)-a(26) added by Ray Chandler, Apr 23 2017
a(27) from Max Alekseyev, May 14 2022

A046420 Prime factors of repunit of length a(n) are all of different lengths.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 11, 14, 17, 19, 23, 31, 34, 37, 38, 41, 43, 47, 49, 51, 53, 57, 59, 62, 67, 69, 71, 73, 74, 79, 83, 85, 86, 89, 93, 94, 97, 101, 103, 106, 109, 113, 115, 118, 119, 129, 134, 137, 139, 141, 142, 146, 149, 151, 157, 159, 163
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Examples

			a(n)=14 -> 11*239*4649*909091 -> (2)(3)(4)(6) all of different lengths.

Links

Crossrefs

Cf. A000042, A004022, A046053.

Programs

  • Mathematica
    Select[Range[60],Length[fi=FactorInteger[(10^#-1)/9]]== Plus@@Last/@fi==Length[Union[IntegerLength/@First/@fi]]&] (* Ray Chandler, Apr 24 2017 *)

Extensions

Offset changed to 1 and more terms added by Ray Chandler, Apr 24 2017

A046414 Repunit of length a(n) has exactly 3 prime factors (counted with multiplicity).

Original entry on oeis.org

13, 31, 37, 38, 67, 73, 83, 97, 101, 103, 113, 127, 149, 163, 191, 227, 241, 263, 313, 331, 373, 379
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Comments

467 <= a(23) <= 857. 857, 1303, 1483, 2267 are terms of this sequence. - Chai Wah Wu, Nov 03 2019

Examples

			a(n)=13 so 1111111111111 = 53*79*265371653.

Links

Crossrefs

Cf. A000042, A001222, A002275, A004022, A046053.

Programs

Extensions

More terms from Robert Gerbicz, Nov 22 2010
More terms from Bo Gyu Jeong, Jun 12 2012

A046430 Lengths of repunits with all prime factors ending with the digit 1.

Original entry on oeis.org

1, 2, 4, 5, 10, 19, 20, 23, 25, 38, 50, 59, 76, 95, 100, 115, 125, 190, 250, 295, 317, 380
Offset: 1

Views

Author

Patrick De Geest, Jul 15 1998

Keywords

Examples

			a(n) = 10 -> 11*41*271*9091 -> 1(1)*4(1)*27(1)*909(1).

Links

Crossrefs

Cf. A000042, A004022, A046053.

Programs

  • Mathematica
    Select[Range[60],Length[fi=FactorInteger[(10^#-1)/9]]==Length[Select[First/@fi,Mod[#,10]==1&]] &](* Ray Chandler, Apr 24 2017 *)

Extensions

a(16)-a(21) from Ray Chandler, Apr 24 2017
a(22) from Max Alekseyev, May 14 2022
Definition corrected by Max Alekseyev, Aug 19 2024

A135981 Number of distinct prime factors of A135972(n).

Original entry on oeis.org

0, 2, 2, 3, 2, 3, 2, 4, 3, 3, 4, 4, 5, 3, 4, 2, 6, 3, 3, 3, 6, 3, 6, 5, 4, 3, 4, 8, 2, 3, 4, 7, 2, 6, 3, 7, 6, 4, 3, 9, 2, 7, 5, 7, 3, 6, 6, 8, 4, 6, 2, 11, 3, 6, 7, 3, 8, 2, 7, 4, 9, 3, 12, 3, 5, 7, 7, 4, 7, 3, 9, 6, 5, 2, 12, 3, 5, 6, 10, 11, 5, 9, 3, 6, 5, 12, 2, 5, 8, 12
Offset: 2

Views

Author

Artur Jasinski, Dec 09 2007

Keywords

Examples

			A135972(3) = 15 = 3*5 which has a(3)=2 distinct prime factors.

Links

Crossrefs

Cf. A046051, A046053, A022307, A135972.

Programs

  • Mathematica
    k = {}; Do[If[ ! PrimeQ[2^n - 1], c = FactorInteger[2^n - 1]; d = Length[c]; AppendTo[k, d]], {n, 1, 100}]; k

Formula

a(n) = A001221(A135972(n)) .

Extensions

Offset set to 2, definition shortened - R. J. Mathar, Oct 01 2009
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