A056105 -id:A056105 - cp's OEIS frontend

A363938 Lexicographically earliest sequence of numbers in a hexagonal spiral such that their neighbors have no common prime factor.

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 8, 17, 15, 19, 12, 23, 25, 29, 10, 31, 14, 27, 16, 21, 37, 33, 26, 41, 22, 35, 43, 47, 18, 49, 24, 53, 39, 20, 51, 55, 28, 59, 61, 32, 45, 34, 67, 57, 40, 63, 71, 36, 65, 38, 73, 79, 30, 83, 77, 46, 89, 69, 91, 58, 81, 85, 87

Offset: 1

Carole Dubois, Jun 29 2023

nonn

			a(11) = 8 because the neighbors of the 11th hexagon are 3, 13, 17, 21, 33, 37, which do not have any common prime divisor with 8.

Carole Dubois, Table of n, a(n) for n = 1..4999
Carole Dubois, Hexagonal spiral for A363938.
Carole Dubois, Scatterplot of A363938(n)
Carole Dubois, Hexagons with spiral

Cf. A000384, A003458, A014105, A056105, A363765.

A131465 a(n)=4n^4-3n^3+2n^2-n+1.

Original entry on oeis.org

1, 3, 47, 259, 861, 2171, 4603, 8667, 14969, 24211, 37191, 54803, 78037, 107979, 145811, 192811, 250353, 319907, 403039, 501411, 616781, 751003, 906027, 1083899, 1286761, 1516851, 1776503, 2068147, 2394309, 2757611, 3160771

Offset: 0

Mohammad K. Azarian, Jul 26 2007

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Cf. A084849, A056109, A056105.

Table[4n^4-3n^3+2n^2-n+1,{n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,3,47,259,861},40] (* Harvey P. Dale, May 27 2017 *)

a(n)=4*n^4-3*n^3+2*n^2-n+1 \\ Charles R Greathouse IV, Oct 21 2022

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Original entry on oeis.org

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Paul Curtz, Jan 22 2016

Trisections:
3,  6,  6,  27, 27, 66,  66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.
3,  3,  18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.
2, 11,  11, 38, 38, 83,  83, ...   (== 2 (mod 9)).
The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

a(n) = (A261327(n+2) + A261327(n-3))/5.
a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.
a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.
a(n) = 3 + A174474(n).
a(2n) + a(2n+1) = A255844(n).
From Colin Barker, Jan 22 2016: (Start)
a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.
a(n) = (n^2 - 4*n + 12)/4 for n even.
a(n) = (n^2 + 2*n + 9)/4 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).
(End)

More terms from Colin Barker, Jan 22 2016

A354947 Number of primes adjacent to prime(n) in a hexagonal spiral of positive integers.

Original entry on oeis.org

2, 2, 0, 2, 1, 1, 0, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0

Offset: 1

Wade Reece Eberly, Sep 23 2022

			The spiral begins
      13--12--11
      /         \
    14   4---3  10
    /   /     \   \
  15   5   1---2   9
    \   \         /
    16   6---7---8
      \
      17--18--19--...
For n=4, prime(4) = 7 in the spiral has a(4) = 2 primes adjacent (2 and 19).

Dimitri Tishchenko, Hexagon Prime Spiral

Cf. A307011, A307013 (spiral coordinates), A056105 (spiral first spoke).

cp's OEIS Frontend

A363938 Lexicographically earliest sequence of numbers in a hexagonal spiral such that their neighbors have no common prime factor.

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A131465 a(n)=4n^4-3n^3+2n^2-n+1.

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A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

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A354947 Number of primes adjacent to prime(n) in a hexagonal spiral of positive integers.

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