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.

A047245 Numbers that are congruent to {1, 2, 3} mod 6.

Original entry on oeis.org

1, 2, 3, 7, 8, 9, 13, 14, 15, 19, 20, 21, 25, 26, 27, 31, 32, 33, 37, 38, 39, 43, 44, 45, 49, 50, 51, 55, 56, 57, 61, 62, 63, 67, 68, 69, 73, 74, 75, 79, 80, 81, 85, 86, 87, 91, 92, 93, 97, 98, 99, 103, 104, 105, 109, 110, 111, 115, 116, 117, 121, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

a(k)^m is a term iff {a(k) is odd and m is a nonnegative integer} or {m is in A004273}. - Jerzy R Borysowicz, May 08 2023

Links

Crossrefs

Cf. A047240, A047244, A047258 (complement).

Programs

Formula

From Johannes W. Meijer, Jun 07 2011: (Start)
a(n) = ceiling(n/3) + ceiling((n-1)/3) + ceiling((n-2)/3) + 3*ceiling((n-3)/3).
G.f.: x*(1+x+x^2+3*x^3)/((x-1)^2*(x^2+x+1)). (End)
a(n) = 3*floor((n-1)/3) + n. - Gary Detlefs, Dec 22 2011
From Wesley Ivan Hurt, Apr 13 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
a(n) = 2*n-3 + ((2*n-3) mod 3). (End)
From Wesley Ivan Hurt, Jun 13 2016: (Start)
a(n) = 2*n - 2 - cos(2*n*Pi/3) + sin(2*n*Pi/3)/sqrt(3).
a(3k) = 6k-3, a(3k-1) = 6k-4, a(3k-2) = 6k-5. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (9-2*sqrt(3))*Pi/36 + log(2+sqrt(3))/(2*sqrt(3)) - log(2)/6. - Amiram Eldar, Dec 14 2021

A383168 Triangle T(n,k) read by rows: For closed chains of identical regular m-gons with connecting inner vertices lying n vertices apart, the n-th row lists the possible m in ascending order; n>=0, 1<=k<=d(8+4n).

Original entry on oeis.org

5, 6, 8, 12, 7, 8, 9, 10, 12, 18, 9, 10, 12, 16, 24, 11, 12, 14, 15, 20, 30, 13, 14, 15, 16, 18, 20, 24, 36, 15, 16, 18, 21, 28, 42, 17, 18, 20, 24, 32, 48, 19, 20, 21, 22, 24, 27, 30, 36, 54, 21, 22, 24, 25, 28, 30, 40, 60, 23, 24, 26, 33, 44, 66
Offset: 1

Views

Author

Manfred Boergens, Apr 18 2025

Keywords

Comments

Consider j identical regular m-gons, assembled into a circular closed chain. Two neighboring polygons share an edge and two vertices, the "inner" one lying in the interior of the chain. The interior is a j-pointed star with equal edges.
n is introduced in order to partition the set of chains into finite subsets. Two neighboring star points are separated by n vertices; there the star has reflex angles. (With n=0, regular polygons are considered as stars with no reflex angles.)
For every m > 4 there exists a chain of m-gons.
A366872 gives the number of row elements.
This sequence is interconnected with A383169. For each n there are finitely many pairs (m,j) for j m-gons building closed chains. m are given by T(n,k) and the corresponding j are given by A383169(n,k).
j = 2 + (8+4n)/(m-4-2n).
m = 4 + 2n + (8+4n)/(j-2).
These two equations allow a computation of T(n,k) and A383169(n,k) from each other, see Formula.

Examples

			Triangle begins:
  5,  6,  8, 12;
  7,  8,  9, 10, 12, 18;
  9, 10, 12, 16, 24;
 11, 12, 14, 15, 20, 30;
 13, 14, 15, 16, 18, 20, 24, 36;
 15, 16, 18, 21, 28, 42;
 17, 18, 20, 24, 32, 48;
 19, 20, 21, 22, 24, 27, 30, 36, 54;
 21, 22, 24, 25, 28, 30, 40, 60;
 23, 24, 26, 33, 44, 66;
 25, 26, 27, 28, 30, 32, 36, 40, 48, 72;
 ...
The third row T(2,.) asserts that regular 9-gons, 10-gons, 12-gons, 16-gons and 24-gons are the only regular polygons which can be assembled to a closed chain with connecting inner vertices lying 2 vertices apart.

Links

Crossrefs

Cf. A047244, A366872, A383169.

Programs

  • Mathematica
    Table[4 + 2*n + Divisors[8 + 4 n], {n, 0, 10}]//Flatten

Formula

T(n,k) = 4+2n + (k-th divisor of 8+4n in ascending order).
T(n,k) = 4+2n + A027750(8+4n,k).
T(n,k) = 4+2n + (8+4n)/(A383169(n,k)-2).
A383169(n,k) = 2 + (8+4n)/(T(n,k)-4-2n).
T(n,1) = 5+2n.
T(n,2) = 6+2n.
T(n,2) = A383169(n,2).
T(n,3) = 7+2n if n=1 mod 3, else = 8+2n.
T(n,3) = A047244(5+n).
T(n,d(8+4n)) = 12+6n (last row elements).
T(n,d(8+4n)-1) = 8+4n (second to last row elements).
T(n,d(8+4n)-2) = (10/3)*(2+n) if n=1 mod 3, else = 3*(2+n) (third last row elements).

A101136 Indices of primes in sequence defined by A(0) = 79, A(n) = 10*A(n-1) - 61 for n > 0.

Original entry on oeis.org

0, 2, 3, 6, 8, 12, 32, 36, 75, 146, 296, 1850, 3456, 3608, 45218
Offset: 1

Views

Author

Klaus Brockhaus and Walter Oberschelp (oberschelp(AT)informatik.rwth-aachen.de), Dec 03 2004

Keywords

Comments

Numbers n such that (650*10^n + 61)/9 is prime.
Numbers n such that digit 7 followed by n >= 0 occurrences of digit 2 followed by digit 9 is prime.
Numbers corresponding to terms <= 296 are certified primes.
a(16) > 10^5. - Robert Price, Oct 01 2015
All a(n) == 0, 2 or 3 mod 6 (cf. A047244). - Robert Israel, Oct 01 2015

Examples

			72222229 is prime, hence 6 is a term.

References

  • Klaus Brockhaus and Walter Oberschelp, Zahlenfolgen mit homogenem Ziffernkern, MNU 59/8 (2006), pp. 462-467.

Links

Crossrefs

Cf. A000533, A002275, A103054.

Programs

  • Magma
    [n: n in [0..3*10^2]| IsPrime((650*10^n+61) div 9)]; // Vincenzo Librandi, Oct 02 2015
  • Maple
    select(t -> isprime((650*10^t + 61)/9), [seq(seq(6*s+i,i=[0,2,3]),s=0..700)]); # Robert Israel, Oct 01 2015
  • Mathematica
    Select[Range[0, 100000], PrimeQ[(650*10^# + 61)/9] &] (* Robert Price, Oct 01 2015 *)
  • PARI
    a=79;for(n=0,1000,if(isprime(a),print1(n,","));a=10*a-61)
  • PARI
    for(n=0,1000,if(isprime((650*10^n+61)/9),print1(n,",")))

Formula

a(n) = A103054(n) - 1.

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(15) from Erik Branger May 01 2013 by Ray Chandler, Apr 30 2015
Showing 1-3 of 3 results.

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

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