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.

Previous Showing 11-16 of 16 results.

A182811 Cyclops-Lucas numbers.

Original entry on oeis.org

64079, 1860498, 4870847, 688846502588399
Offset: 1

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Author

Omar E. Pol, Dec 20 2010

Keywords

Comments

a(4) = 688846502588399 is the only known Cyclops-Lucas prime.
It seems likely that these four are the only terms. There are no further terms below Lucas(10^7), and that number in decimal contains 208435 zeros (with ~208988 expected assuming normality), whereas a member of this sequence must have only 1. - D. S. McNeil, Dec 21 2010
This sequence is similar to A182809 in the sense that both have four positive terms and the only known prime is also the largest known term. - Omar E. Pol, Dec 21 2010
Indices in A000032 are 23, 30, 32, 71. - Michel Marcus and Omar E. Pol, Feb 18 2018

Examples

			a(1) = 64079 is in the sequence because 64079 is a Lucas number and it is also a cyclops number.

Links

Crossrefs

Intersection of A000032 and A134808.
Cf. A005479, A134809, A182809.

Programs

  • Mathematica
    (* First run the program given for A134808 *) Select[LucasL[Range[10^3]], cyclopsQ] (* Alonso del Arte, Dec 20 2010 *)
Select[LucasL[Range[500]],OddQ[IntegerLength[#]]&&DigitCount[#,10,0]==1&&IntegerDigits[#][[(IntegerLength[#]+1)/2]]==0&] (* Harvey P. Dale, Jul 01 2017 *)

Formula

Intersection of A000032 and A134808.

A201011 Primes that are Lucas primes, or that can be written as the quotient of Lucas numbers.

Original entry on oeis.org

2, 3, 7, 11, 19, 23, 29, 31, 41, 47, 107, 199, 211, 281, 521, 1103, 2161, 2207, 2521, 3571, 5779, 9349, 9901, 14503, 90481, 103681, 3010349, 11128427, 29134601, 54018521, 261399601, 370248451, 599786069, 6643838879, 10745088481, 10749957121, 10783342081
Offset: 1

Views

Author

Arkadiusz Wesolowski, Jan 08 2013

Keywords

Examples

			23 is in the sequence because it is prime and Lucas(12)/(Lucas(0)*Lucas(4)) = 23.

Links

Crossrefs

Cf. A000032, A200381, A200995, A201010, A201012. Supersequence of A005479. Subsequence of A178762.

Extensions

261399601 inserted by Arkadiusz Wesolowski, Feb 05 2013

A307499 The number of primes between two consecutive prime Lucas numbers, bounds excluded.

Original entry on oeis.org

0, 1, 0, 4, 4, 30, 51, 230, 170, 657, 216347, 3009722, 16603784, 288244979, 4566061654, 192922096576, 20592039889787, 854140717540139, 7734073644382760578105
Offset: 1

Views

Author

Hauke Löffler, Jul 24 2019

Keywords

Examples

			a(0): between the first two prime Lucas numbers (2,3) there are 0 primes.
a(3): between 11 and 29 there are 4 primes (13, 17, 19, 23).

Links

Crossrefs

Cf A005479, A134850, A176559.

Programs

  • Mathematica
    Differences@ PrimePi@ Select[LucasL@ Range[0, 70], PrimeQ] - 1 (* Giovanni Resta, Jul 28 2019 *)
  • SageMath
    # uses[A005479]
def count_primes_between(a, b):
    return len(prime_range(a+1, b))
[count_primes_between(A005479[i], A005479[i+1]) for i in range(len(A005479)-1)]

Extensions

a(14)-a(18) from Giovanni Resta, Jul 28 2019
a(19) using Kim Walisch's primecount, from Amiram Eldar, May 14 2023

A176412 Concatenation of prime Lucas numbers written in base 2.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1
Offset: 1

Views

Author

Vincenzo Librandi, Apr 17 2010

Keywords

Comments

2=10; 3=11; 7=111; 11=1001; 29=11101; 47=101111; 199=11000111; 521=1000001001; 2207=100010011111; 3571=110111110011; 9349=10010010000101; etc.

Crossrefs

Cf. A005479 (prime Lucas numbers).

Extensions

Edited and extended by Charles R Greathouse IV, Apr 25 2010

A194086 Second-smallest prime factor of the n-th Lucas number (beginning with 2), if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 3, 1, 1, 19, 41, 1, 7, 1, 281, 11, 1, 1, 3, 1, 2161, 29, 43, 461, 47, 101, 90481, 19, 14503, 19489, 3, 1, 4481, 199, 67, 29, 7, 1, 29134601, 79, 1601, 1, 3, 144481, 263, 11, 4969, 1, 769, 599786069, 41, 919, 103, 1, 3, 199, 10745088481, 229
Offset: 0

Views

Author

Jonathan Vos Post, Aug 14 2011

Keywords

Examples

			a(3) = 2 because the 3rd of the Lucas numbers (beginning at 2) is A000032(3) = 4 = 2^2, hence depending on whether one means by second-smallest prime factor (i.e., distinct or not, with multiplicity or not) a(3) would be either 2 or 1.
a(10) is unambiguously 41, because L(10) = 123 = 3 * 41, and 41 is the second-smallest prime factor, with no issues of multiplicity or distinctness.

Crossrefs

Cf. A000032, A000040, A001606, A005479, A193615.

Programs

  • Mathematica
    Table[f = FactorInteger[LucasL[n]]; If[Length[f] > 1, f[[2, 1]], If[Length[f] == 1 && f[[1, 2]] > 1, f[[1, 1]], 1]], {n, 0, 70}] (* T. D. Noe, Aug 15 2011 *)

A373920 Isolated prime Lucas numbers.

Original entry on oeis.org

2, 47, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521, 59242995313457729780510823767354730798286848921481374874264534705573628371
Offset: 1

Views

Author

Marc Groz, Jun 22 2024

Keywords

Examples

			47 is a term because it is the 6th prime Lucas number (per A005479) and is an isolated prime (per A007510).

Crossrefs

Intersection of A005479 and A007510.
Cf. A000032.

Extensions

More terms from Michael S. Branicky, Jun 23 2024
Previous Showing 11-16 of 16 results.

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