A242488 -id:A242488 - cp's OEIS frontend

A243851 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

1, 1, 1, 1, 2, 1, 2, 4, 1, 2, 4, 5, 5, 1, 2, 4, 5, 7, 5, 7, 2, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 7, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 13, 7, 13, 19, 16, 5, 11, 8, 1, 2, 4, 5, 7, 5, 8, 7, 11, 11, 3, 13, 7, 13, 19, 10, 16, 5, 11, 23, 8, 26, 20, 23, 6, 26, 7

Offset: 1

Clark Kimberling, Jun 12 2014

Decree that (row 1) = (1) and (row 2) = (3,2). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 3/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.

			First 6 rows of the array of rationals:
1/1
3/1 ... 2/1
4/1 ... 3/2
5/1 ... 5/2 ... 3/4
6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
The denominators, by rows:  1,1,1,1,2,1,2,4,1,2,4,5,5,1,2,4,5,7,5,7,2.

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A243852, A243853, A242488.

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Denominator[v] (* A243851 *)
Numerator[v]   (* A243852 *)
Table[Length[g[n]], {n, 1, z}] (* A243853 *)

A243852 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

Original entry on oeis.org

1, 3, 2, 4, 3, 5, 5, 3, 6, 7, 7, 6, 3, 7, 9, 11, 11, 12, 8, 6, 1, 8, 11, 15, 16, 19, 13, 15, 13, 15, 12, 2, 3, 9, 13, 19, 21, 26, 18, 23, 20, 26, 23, 5, 21, 10, 15, 21, 15, 4, 6, 3, 10, 15, 23, 26, 33, 23, 31, 27, 37, 34, 8, 34, 17, 28, 40, 21, 31, 9, 17, 33

Offset: 1

Clark Kimberling, Jun 12 2014

Decree that (row 1) = (1) and (row 2) = (3,2). For n >= 4, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 3/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.

			First 6 rows of the array of rationals:
1/1
3/1 ... 2/1
4/1 ... 3/2
5/1 ... 5/2 ... 3/4
6/1 ... 7/2 ... 7/4 ... 6/5 ... 3/5
7/1 ... 9/2 ... 11/4 .. 11/5 .. 12/7 .. 8/5 .. 6/7 .. 1/2
The numerators, by rows:  1,3,2,4,3,5,5,3,6,7,7,6,3,7,9,11,11,12,8,6,1.

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A243851, A243853, A242488.

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 3/x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Denominator[v] (* A243851 *)
Numerator[v]   (* A243852 *)
Table[Length[g[n]], {n, 1, z}] (* A243853 *)

A243854 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 5, 1, 3, 5, 3, 1, 3, 5, 9, 7, 3, 7, 1, 3, 5, 9, 7, 3, 5, 7, 7, 5, 2, 1, 3, 5, 9, 7, 3, 5, 7, 11, 7, 5, 2, 19, 29, 19, 13, 9, 1, 3, 5, 9, 7, 3, 5, 7, 11, 7, 5, 2, 19, 29, 19, 13, 11, 17, 9, 9, 17, 11, 13, 19, 6, 4, 5, 1, 3, 5, 9, 7, 3, 5

Offset: 1

Clark Kimberling, Jun 12 2014

Decree that (row 1) = (1). For n >= 2, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 4/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.

			First 6 rows of the array of rationals:
1/1
4/1 ... 2/1
5/1 ... 3/1
6/1 ... 4/3 ... 4/5
7/1 ... 7/3 ... 9/5 ... 2/3
8/1 ... 10/3 ... 14/5 .. 20/9 .. 12/7 .. 5/3 .. 4/7
The denominators, by rows:  1,1,1,1,1,1,3,5,1,3,5,3,1,3,5,9,7,3,7.

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A243855, A243856, A242488, A243848.

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4/x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Denominator[v] (* A243854 *)
Numerator[v]   (* A243855 *)
Table[Length[g[n]], {n, 1, z}] (* A243856 *)

A243855 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

Original entry on oeis.org

1, 4, 2, 5, 3, 6, 4, 4, 7, 7, 9, 2, 8, 10, 14, 20, 12, 5, 4, 9, 13, 19, 29, 19, 8, 12, 11, 10, 6, 1, 10, 16, 24, 38, 26, 11, 17, 18, 28, 17, 11, 3, 28, 36, 20, 12, 4, 11, 19, 29, 47, 33, 14, 22, 25, 39, 24, 16, 5, 47, 65, 39, 25, 20, 28, 14, 13, 20, 12, 14

Offset: 1

Clark Kimberling, Jun 12 2014

Decree that (row 1) = (1). For n >= 2, row n consists of numbers in decreasing order generated as follows: x+1 for each x in row n-1 together with 4/x for each x in row n-1, and duplicates are rejected as they occur. Every positive rational number occurs exactly once in the resulting array.

			First 6 rows of the array of rationals:
1/1
4/1 ... 2/1
5/1 ... 3/1
6/1 ... 4/3 ... 4/5
7/1 ... 7/3 ... 9/5 ... 2/3
8/1 ... 10/3 ... 14/5 .. 20/9 .. 12/7 .. 5/3 .. 4/7
The numerators, by rows:  1,4,2,5,3,6,4,4,,7,7,9,2,8,10,14,20,12,5,4.

Clark Kimberling, Table of n, a(n) for n = 1..3000

Cf. A243854, A243856, A242488, A243848.

z = 12; g[1] = {1}; f1[x_] := x + 1; f2[x_] := 4/x; h[1] = g[1];
b[n_] := b[n] = DeleteDuplicates[Union[f1[g[n - 1]], f2[g[n - 1]]]];
h[n_] := h[n] = Union[h[n - 1], g[n - 1]];
g[n_] := g[n] = Complement [b[n], Intersection[b[n], h[n]]]
u = Table[Reverse[g[n]], {n, 1, z}]; v = Flatten[u];
Denominator[v] (* A243854 *)
Numerator[v]   (* A243855 *)
Table[Length[g[n]], {n, 1, z}] (* A243856 *)

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

Original entry on oeis.org

0, 1, 2, 4, 5, 22, 3, 8, 14, 19, 140, 7, 10, 24, 41, 58, 265, 707, 6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362, 11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991, 16, 27, 59, 70, 102, 113, 317, 500, 586, 1048, 2951, 3424, 4972, 8240, 12658, 83834, 686210, 1306066

Offset: 1

Andrew Howroyd, Dec 22 2024

For any prime p, there are finitely many x such that x^2 + 2 has p as its greatest prime factor.

			Irregular triangle begins:
   p | {k}
-----+------------------
   2 | {0}
   3 | {1, 2, 4, 5, 22}
  11 | {3, 8, 14, 19, 140}
  17 | {7, 10, 24, 41, 58, 265, 707}
  19 | {6, 13, 25, 32, 44, 63, 146, 184, 602, 3407, 21362}
  41 | {11, 30, 52, 71, 112, 194, 298, 481, 503, 2695, 3433, 4991}
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..915 (first 21 rows for primes up to 193)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for rows 2..21 (=914 terms) of this sequence).

Cf. A033203, A379351, A379352 (first terms), A185397 (last terms), A379349 (row lengths).

Cf. A223701, A223702, A242488.

A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

2, 10, 108, 235, 1201, 390050, 314766, 4035, 364384, 50411, 25955045, 5254864, 236558593, 16958526, 20388056, 177544434, 492981885, 2275400230, 256347346, 384902923486, 324850200677887

Offset: 1

Charles R Greathouse IV, Feb 21 2011

nonn

For any prime p, there are finitely many x such that x^2-2 has p as its largest prime factor.

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, Home Page (gives all 537 numbers x such that x^2-2 has no prime factor greater than 199)

Cf. A242488, A379348.

Equivalents for other polynomials: A175607 (x^2 - 1), A145606 (x^2 + x), A185389 (x^2 + 1).

a(21) added by Andrew Howroyd, Dec 22 2024

A379348 Number of positive integers of the form k^2 - 2 whose greatest prime factor is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

Original entry on oeis.org

1, 3, 4, 6, 5, 8, 10, 10, 10, 14, 20, 22, 30, 30, 37, 42, 43, 48, 49, 64, 80

Offset: 1

Andrew Howroyd, Dec 22 2024

See A242488 for additional information.

			Table showing n, p = A038873(n) and a(n):
   1    2    1
   2    7    3
   3   17    4
   4   23    6
   5   31    5
   6   41    8
   7   47   10
   8   71   10
   9   73   10
  10   79   14
  ...

Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.

Row lengths of A242488.

Cf. A038873, A185396, A379349.

cp's OEIS Frontend

A243851 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

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A243852 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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A243854 Irregular triangular array of denominators of the positive rational numbers ordered as in Comments.

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Programs

Mathematica

A243855 Irregular triangular array of numerators of the positive rational numbers ordered as in Comments.

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Programs

Mathematica

A379350 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 + 2 is A033203(n), the n-th prime not congruent to 5 or 7 mod 8.

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A185396 Largest number x such that the greatest prime factor of x^2-2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A379348 Number of positive integers of the form k^2 - 2 whose greatest prime factor is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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