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.

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

A069882 Numbers n such that n and 2n-1 are both palindromes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 66, 666, 6666, 66666, 666666, 6666666, 66666666, 666666666, 6666666666, 66666666666, 666666666666, 6666666666666, 66666666666666, 666666666666666, 6666666666666666, 66666666666666666, 666666666666666666
Offset: 1

Views

Author

Amarnath Murthy, Apr 30 2002

Keywords

Comments

From Chai Wah Wu, Jul 20 2020: (Start)
Theorem: a(n) = 2*(10^(n-5)-1)/3 for n > 5.
Proof: clearly 2*(10^m-1)/3 are terms of this sequence. Next we show that all terms > 10 are of the form 2*(10^m-1)/3. Let k > 10 be a term of the sequence. Let x be the first digit (and thus also the last digit) of k. If x <> 6 then it is easy to show that the first and last digit of 2k-1 will not be the same. Thus x = 6. Let the digits of k be written as 6y****y6. Similarly if y <> 6 then again the second digit of 2k-1 will not be the same as the second to last digit of 2k-1. Continuing in this manner, this shows that k written in decimal is a sequence of 6's.
(End)

Examples

			66 is a member as 2*66 - 1 = 131 is also a palindrome.

Links

Crossrefs

Cf. A002280, A069881, A185127, A259050.

Formula

From Chai Wah Wu, Jul 20 2020: (Start)
a(n) = 2*(10^(n-5)-1)/3 for n > 5.
a(n) = 11*a(n-1) - 10*a(n-2) for n > 7.
G.f.: x*(50*x^6 - 9*x^5 - 9*x^4 - 9*x^3 - 9*x^2 - 9*x + 1)/((x - 1)*(10*x - 1)).
(End)

Extensions

More terms from Hans Havermann, Jul 06 2002

A326929 Numbers whose divisors and arithmetic mean of divisors are palindromic.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 22, 131, 262, 13331, 26662, 1333331, 2666662
Offset: 1

Views

Author

Ivan N. Ianakiev, Oct 22 2019

Keywords

Comments

a(14) is greater than 10^18 and at most (10^94-1)*(4/3)-1. - Charles R Greathouse IV, Oct 28 2019
(4*10^A259050(n)-7)/3 and (8*10^A259050(n)-14)/3 are terms. Conjecture: all terms > 10 are of these forms. - Chai Wah Wu, Nov 17 2019

Examples

			The divisors of 2666662 are {1,2,1333331,2666662} which are all palindromic. Their arithmetic mean is 999999 and is also palindromic. Therefore, 2666662 is in the sequence.

Crossrefs

Subsequence of A062687 and hence of A002113.
Subsequence of A003601.
Cf. A259050.

Programs

  • Mathematica
    palQ[n_]:=ToString[n]==StringReverse[ToString[n]];
fQ[n_]:=palQ[Mean[Divisors[n]]]&&Union[palQ/@Divisors[n]]=={True};
Select[Range[2666662],fQ]
Showing 1-3 of 3 results.

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

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