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 1-4 of 4 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

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Author

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

Crossrefs

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

A097166 Expansion of g.f. (1+2*x)/((1-x)*(1-10*x)).

Original entry on oeis.org

1, 13, 133, 1333, 13333, 133333, 1333333, 13333333, 133333333, 1333333333, 13333333333, 133333333333, 1333333333333, 13333333333333, 133333333333333, 1333333333333333, 13333333333333333, 133333333333333333, 1333333333333333333, 13333333333333333333, 133333333333333333333
Offset: 0

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Author

Paul Barry, Jul 30 2004

Keywords

Comments

Partial sums of (1+2*x)/(1-10*x) = {1, 12, 120, 1200, ...}.
Second binomial transform of A082365.
These terms are the x's of A070152 and the corresponding y's are A350995 (see formula and examples). - Bernard Schott, Feb 15 2022

Examples

			a(0) = (4-1)/3 = 1 and Sum_{j=1..5} = 15.
a(1) = (40-1)/3 = 13 and Sum_{j=13..53} = 1353.
a(2) = (400-1)/3 = 133 and Sum_{j=133..533} = 133533.
		

Crossrefs

Cf. A056698 (index of primes), A082365, A097169, A309907 (squares of this).

Programs

  • Magma
    [(4*10^n-1)/3 : n in [0..20]]; // Vincenzo Librandi, Nov 01 2011
    
  • Maple
    a:= n-> parse(cat(1, 3$n)):
    seq(a(n), n=0..18);  # Alois P. Heinz, Aug 23 2019
  • Mathematica
    NestList[10#+3&,1,20] (* Harvey P. Dale, Jan 22 2014 *)
  • Python
    [(4*10**n-1)//3 for n in range(25)] # Gennady Eremin, Mar 04 2022

Formula

a(n) = (4*10^n - 1)/3.
a(n) = A097169(2*n).
a(n) = 10*a(n-1) + 3, n>0. a(n) = 11*a(n-1) - 10*a(n-2), n>1. - Vincenzo Librandi, Nov 01 2011
A350994(n) = Sum_{j=a(n)..A350995(n)} = a(n).A350995(n) where "." means concatenation. - Bernard Schott, Jan 28 2022
From Elmo R. Oliveira, Apr 29 2025: (Start)
E.g.f.: exp(x)*(4*exp(9*x) - 1)/3.
a(n) = A198970(n)/3. (End)

A093671 Primes of the form 10^k + 3*R_k, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

13, 1333333333333333, 133333333333333333333333333333333333333333, 133333333333333333333333333333333333333333333333333333333333333333333333333333333333
Offset: 1

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Author

Rick L. Shepherd, Apr 08 2004

Keywords

Crossrefs

Cf. A002275 (repunits), A056698 (corresponding k).

Programs

  • Magma
    [a: n in [0..100] | IsPrime(a) where a is ((4*10^n-1) div 3)]; // Vincenzo Librandi, Apr 06 2019
  • Mathematica
    Select[Table[FromDigits[PadRight[{1}, n, 3]], {n, 500}], PrimeQ] (* Vincenzo Librandi, Apr 06 2019 *)

A105427 Numbers n such that the near-repdigit number consisting of a 1 followed by n 3's (i.e., of form 1333...33) is composite.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75
Offset: 1

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Author

Lekraj Beedassy, Apr 08 2005

Keywords

Comments

Complement of A056698.

Crossrefs

Programs

  • Mathematica
    Select[Range[100],CompositeQ[FromDigits[PadRight[{1},#,3]]]&]-1 (* Harvey P. Dale, Jul 23 2014 *)
  • PARI
    isok(n) = ! isprime(10^n+(10^n-1)/3) \\ Michel Marcus, Jul 28 2013
Showing 1-4 of 4 results.