A188552 Prime numbers at locations of angle turns in pentagonal spiral.

2, 3, 5, 7, 11, 17, 23, 31, 59, 71, 83, 97, 127, 179, 199, 241, 263, 311, 337, 419, 449, 479, 577, 647, 683, 839, 881, 967, 1103, 1151, 1249, 1511, 1567, 2111, 2243, 2311, 2591, 2663, 2887, 2963, 3041, 3119, 3361, 3527, 3697, 4049, 4139, 4231, 4703, 4801, 4999, 5099

Offset

1,1

Comments

2, 5, and primes in A056220 (lower vertical in pdf) and primes in A142463 (upper vertical). [Joerg Arndt, Apr 13 2011]

The link gives an illustration with three figures: Figure 1 contains the prime numbers at locations of angle turns in a pentagonal spiral; Figure 2 contains the prime numbers in a pentagonal spiral; Figure 3 shows a variety of sequences that are associated with the numbers of the lines and diagonals in the pentagonal spiral. For example, the sequence A033537 given by the formula n(2n+5) generates the sequence {0, 7, 18, 33, 52, 75, ... } and the corresponding line in the spiral is { 7, 18, 33, 52, 75, ... }.

Links

Table of n, a(n) for n=1..52.

Michel Lagneau, Illustration of the numbers in the pentagonal spiral

Example

The pentagonal spiral's changes of direction (vertices) occur at the primes 2, 3, 5, 7, 11, 17, 23 ...

Maple

with(numtheory): T:=array(1..300):k:=1:for n from 1 to 50 do:x1:= 2*n^2 -1:
  T[k]:=x1: x2:= (n+1)*(2*n-1): T[k+1]:=x2:x3:=2*n^2+2*n-1 : T[k+2]:=x3:x4:= 2*n*(n+1):
  T[k+3]:=x4:x5:=n*(2*n+3): T[k+4]:=x5:k:=k+5:od: for p from 1 to 250 do:z:= T[p]:if
  type(z, prime)= true then printf(`%d, `, z):else fi:od:

Crossrefs

Cf. A188551.

Sequence in context: A005105 A086566 A235213 * A104892 A065436 A068523

Adjacent sequences: A188549 A188550 A188551 * A188553 A188554 A188555

Keyword

nonn

Author

Michel Lagneau, Apr 04 2011

Status

approved