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.

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

A093170 Primes of the form 60*R_k + 7, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

7, 67, 666667, 66666667, 666666667, 66666666667, 66666666666666666667, 66666666666666666666667, 66666666666666666666666666666666666666667, 666666666666666666666666666666666666666666666666666666666666667
Offset: 1

Views

Author

Rick L. Shepherd, Mar 26 2004

Keywords

Comments

Primes of the form (2*10^k + 1)/3. - Vincenzo Librandi, Nov 16 2010
Occur in the factorization of some of the numbers of the form 13...3 not in A093671, cf. second Kamada link. - M. F. Hasler, Sep 14 2014

Links

Crossrefs

Cf. A002275, A056657 (corresponding k), A093671, A096507.

Programs

  • Maple
    A093170:=n->`if`(isprime((2*10^n+1)/3),(2*10^n+1)/3,NULL): seq(A093170(n), n=1..70); # Wesley Ivan Hurt, Sep 14 2014
  • Mathematica
    Select[Table[FromDigits[PadLeft[{7},n,6]],{n,70}],PrimeQ] (* Harvey P. Dale, Jan 26 2013 *)

Formula

a(n) = (20*10^A056657(n)+1)/3 = (2*10^A096507(n)+1)/3.

Extensions

Edited by Ray Chandler, Feb 23 2012

A056698 Numbers k such that 10^k + 3*R_k is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 15, 41, 83, 95, 341, 551, 669, 989, 1223, 6923, 103703
Offset: 1

Views

Author

Robert G. Wilson v, Aug 10 2000

Keywords

Comments

Also numbers k such that (4*10^k-1)/3 is prime.
a(13) > 850000 (from Kamada data). - Robert Price, Oct 19 2014

Links

Crossrefs

Cf. A002275, A093671, A097166, A259050.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 10^n + 3*(10^n-1)/9], Print[n]], {n, 0, 30470}]
  • PARI
    for(k=1,1500,if(ispseudoprime(4*(10^k-1)/3+1),print1(k, ", "))) \\ Hugo Pfoertner, Jul 22 2020

Extensions

a(12) from Kamada data by Robert Price, Oct 19 2014

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

Views

Author

Lekraj Beedassy, Apr 08 2005

Keywords

Comments

Complement of A056698.

Links

Crossrefs

Cf. A051200, A089017, A093671.

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

A276827 Primes p such that the greatest prime factor of 3*p+1 is at most 5.

Original entry on oeis.org

3, 5, 13, 53, 83, 853, 2083, 3413, 5333, 85333, 208333, 218453, 341333, 3495253, 5461333, 8533333, 13981013, 83333333, 853333333, 22369621333, 218453333333, 341333333333, 2236962133333, 3665038759253, 53333333333333, 91625968981333, 203450520833333, 1333333333333333
Offset: 1

Views

Author

Robert Israel, Sep 19 2016

Keywords

Comments

Prime(i) such that A087273(i) <= 5.

Links

Crossrefs

Cf. A087273.
Contains A093671, A093674, and A093676.

Programs

  • Maple
    N = 10^20: # to get all terms <= N
Res:= {}:
for a from 0 to ilog2(floor((3*N+1)/5)) do
  twoa:= 2^a;
  for b from (a mod 2) by 2 do
    p:= (twoa*5^b-1)/3;
    if p > N then break fi;
    if isprime(p) then
      Res:= Res union {p};
    fi
od od:
sort(convert(Res,list));
  • Mathematica
    Select[Prime@ Range[10^6], FactorInteger[3 # + 1][[-1, 1]] <= 5 &] (* Michael De Vlieger, Sep 19 2016 *)
  • PARI
    list(lim)=my(v=List(),s,t); lim=lim\1*3 + 1; for(i=0,logint(lim\2,5), t=if(i%2,2,4)*5^i; while(t<=lim, if(isprime(p=t\3), listput(v,p)); t<<=2)); Set(v) \\ Charles R Greathouse IV, Sep 19 2016
Showing 1-5 of 5 results.

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

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