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 1-3 of 3 results.

A051200 Except for initial term, primes of form "n 3's followed by 1".

Original entry on oeis.org

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

"A remarkable pattern that is entirely accidental and leads nowhere" - M. Gardner, referring to the first 8 terms.
a(2)*a(3)*a(4) = 34179391, a Zeisel number (A051015) with coefficients (10,21).
a(2)*a(3)*a(4)*a(5) = 1139233281421, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(6) = 379741768929343351, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(7) = 1265805010367017001532181, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(8) = 42193497392022209194699696424911, a Zeisel number with coefficients (10,21).
Besides first 3, primes of the form (10^n-7)/3, n>1. See A123568. - Vincenzo Librandi, Aug 06 2010
The integer lengths of the terms of the sequence are 1, 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, etc. - Harvey P. Dale, Dec 01 2018

References

  • Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 194.
  • W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb, Warsaw, 1964; Problem 88 [in English: 200 Problems from the Elementary Theory of Numbers]
  • W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, pp. 8, 56-57.
  • F. Smarandache, Properties of numbers, University of Craiova, 1973

Links

Crossrefs

Cf. A055520, A089017, A089018, A093671, A056698, A105427, A105428, A033175, A123568.

Programs

  • Mathematica
    Join[{3},Select[Rest[FromDigits/@Table[PadLeft[{1},n,3], {n,50}]], PrimeQ]]  (* Harvey P. Dale, Apr 20 2011 *)

Formula

Union of 3 and A123568.

Extensions

More terms from James Sellers
Cross reference added by Harvey P. Dale, May 21 2014

A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1732, 1918, 8855, 11245, 11960, 12130, 18533, 22718, 23365, 24253, 24549, 25324, 30178, 53718, 380976, 424861, 563535, 666903
Offset: 1

Views

Author

Labos Elemer, Jul 10 2000

Keywords

Comments

Also numbers k such that (10^k-7)/3 is prime.
Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.
The history of the discovery of these numbers may be as follows: a(1)-a(7), Makowski; a(8)-a(18), Caldwell; a(19), Earls; a(20)-a(31), Kamada. (Corrections to this account will be welcomed.)
Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos
No more than 14 consecutive exponents can provide primes because for exponents 15m+2, 16m+9, 18m+12, 22m+21, terms are divisible by 31, 17, 19, 23 respectively. Here 7 of possible 14 is realized. - Labos Elemer, Jan 19 2005
(10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - Seiichi Manyama, Nov 05 2016

References

  • C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
  • S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
  • W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.

Links

Crossrefs

Cf. A051200, A033175, A055520.

Programs

  • Mathematica
    Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
One may run the prime certificate program as follows <True]}, {n, 1, 16}] (* Labos Elemer *)
  • PARI
    for(n=1,2000, if(isprime((10^n-7)/3),print(n)))

Formula

a(n) = A055520(n) + 1.

Extensions

Corrected and extended by Jason Earls, Sep 22 2001
a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.
a(32)-a(33) from Leonid Durman, Jan 09-10 2012
a(34)-a(35) from Kamada data by Tyler Busby, Apr 14 2024

A089017 n for which the number consisting of a string of n 3's and a terminal 1 is not prime.

Original entry on oeis.org

8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82
Offset: 1

Views

Author

Lekraj Beedassy, Nov 04 2003

Keywords

Comments

Complement of A055520(n)=A055557(n) - 1. The first n for which {10^(n+1) - 7}/3 is composite is thus n=8,corresponding to 333333331=17*19607843.

Crossrefs

Cf. A055520, A089018.

Programs

  • Mathematica
    Select[Range[2,90],!PrimeQ[FromDigits[PadLeft[{1},#,3]]]&]-1 (* Harvey P. Dale, Jun 19 2012 *)
  • PARI
    is(n)=ispseudoprime((10^(n+1)-7)/3) \\ Charles R Greathouse IV, Oct 23 2013
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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