A192179 -id:A192179 - cp's OEIS frontend

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203, 125, 57, 22

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

A permutation of natural numbers >= 2.
The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - Peter Kagey, Jun 03 2019, based on comments by Jamie Morken and discussion with Tom Hanlon.
The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - Tom Hanlon, Jun 08 2019

			Array begins:
   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
Arrays of integers grouped into rows by various criteria:
by greatest prime factor: A125624,
by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
by number of distinct prime factors: A125666,
by number of prime factors counted with multiplicity: A078840,
by prime signature: A095904,
by ordered prime signature: A096153,
by number of divisors: A119586,
by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
by distance to next prime: A192179.
Cf. A002110, A005867, A038110, A038111.

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
(* second program: *)
rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-François Alcover, Mar 08 2016 *)

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

A192175 Array of primes determined by distance to next prime, by antidiagonals.

Original entry on oeis.org

2, 3, 7, 5, 13, 23, 11, 19, 31, 89, 17, 37, 47, 359, 139, 29, 43, 53, 389, 181, 199, 41, 67, 61, 401, 241, 211, 113, 59, 79, 73, 449, 283, 467, 293, 1831, 71, 97, 83, 479, 337, 509, 317, 1933, 523, 101, 103, 131, 491, 409, 619, 773, 2113, 1069, 887, 107

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  primes p such that p+1 or p+2 is a prime.
Row r>1:  primes p such that the least h for which p+2h is prime is r.
Rows 1-7:  A124588, A023200, A031924, A031926, A031928, A031932, A031924.

			Northwest corner:
  2.....3.....5.....11....17....29....41
  7.....13....19....37....43....67....79
  23....31....47....53....61....73....83
  89....359...389...401...449...479...491
  139...181...241...283...337...409...421
For example, 31 is in row 3 because 31+2*3 is a prime, unlike 31+2*1 and 31+2*2.  Every prime occurs exactly once.  For each row, it is not known whether it is finite.

Cf. A192176, A192177, A192178, A192179.

z = 5000; (* z=number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1,
      PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 16}]; TableForm[Array[Prime[row[#]] &, {10}]] (* A192175 array *)
Flatten[Table[ Prime[row[k][[n - k + 1]]], {n, 1, 11}, {k, 1, n}]] (* A192175 sequence *)
(* Peter J. C. Moses, Jun 20 2011 *)

A192176 Index array for A192175 (distance up to next prime), by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 8, 11, 24, 7, 12, 15, 72, 34, 10, 14, 16, 77, 42, 46, 13, 19, 18, 79, 53, 47, 30, 17, 22, 21, 87, 61, 91, 62, 282, 20, 25, 23, 92, 68, 97, 66, 295, 99, 26, 27, 32, 94, 80, 114, 137, 319, 180, 154, 28, 29, 36, 124, 82, 121, 146, 331, 205, 259, 189

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1: numbers k such that p + 1 or p + 2 is a prime,
where p = (k-th prime).
Row r > 1: numbers k such that if p = (k-th prime) then r = (least h for which p + 2 h) is a prime.
Every positive integer occurs exactly once, so that as a sequence, A192176 is a permutation of the positive integers.

			Northwest corner:
1....2....3....5....7....10....13
4....6....8....12...14...19....22
9....11...15...16...18...21....23
24...72...77...79...87...92....94
34...42...53...61...68...80....82
...
These are the index numbers of the primes displayed in the Example at A192175; e.g., in that display, the top row begins with 2,3,5,11,17,29,41.

Cf. A192175, A192177, A192178, A192179.

z = 5000; (* z = number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[1 + Prime[#1]] || PrimeQ[2 + Prime[#1]]} &, {z}], {, True}]; Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[2 x + Prime[#1]]} &, {z}], {, True}],Flatten[Array[row, {x - 1}]]], {x, 2, 16}];
TableForm[Array[row, {16}]]   (* A192176 array *)
Flatten[Table[row[k][[n - k + 1]], {n, 1, 11}, {k, 1, n}]]   (* A192176 sequence *)
(* by Peter J. C. Moses, Jun 20 2011 *)

A192177 Array determined by distance down to nearest prime.

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 4, 9, 16, 11, 6, 13, 22, 17, 28, 8, 15, 26, 23, 36, 29, 12, 19, 34, 27, 52, 37, 96, 14, 21, 40, 35, 58, 53, 120, 97, 18, 25, 46, 41, 66, 59, 146, 121, 122, 20, 31, 50, 47, 78, 67, 188, 147, 148, 123, 24, 33, 56, 51, 88, 79, 206, 189, 190, 149, 11

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  numbers k such that k = 1 or k = 2 or k - 1 is a prime.
Row r > 1: numbers k such that k - r is a prime and k - q is not, for q = 1, 2, ..., r - 1.
Every positive integer occurs exactly once, so that as a sequence, A192177 is a permutation of the positive integers.
For r>1, the numbers in row r have the parity of r-1; e.g., the numbers in row 2 are odd.

			Northwest corner:
1....2....3....4....6....8
5....7....9....13...15...19
10...16...22...26...34...40
11...17...23...27...35...41
28...36...52...58...66...78
...
For example, 16 is in row 3 because 16 - 3 is prime, unlike 16 - 1 and 16 - 2.

Cf. A192175, A192176, A192178, A192179.

z = 5000; (* z = number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[#1 - 1] || #1 == 1 || #1 == 2} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@  Cases[Array[{#1, PrimeQ[#1 - x]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 10}];
TableForm[Array[row, {10}]]  (* A192177 array *)
Flatten[Table[row[k][[n - k + 1]], {n, 1, 11}, {k, 1,
   n}]]  (* A192177 sequence *)
(* by Peter J. C. Moses, Jun 24 2011 *)

A192178 Array by distance to nearest prime, by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 26, 4, 9, 34, 23, 6, 11, 50, 37, 118, 8, 13, 56, 47, 122, 53, 10, 15, 64, 67, 144, 89, 120, 12, 17, 76, 79, 186, 119, 300, 409, 14, 19, 86, 83, 204, 121, 324, 479, 532, 16, 21, 92, 93, 206, 157, 530, 531, 896, 293, 18, 25, 94, 97, 216, 173, 534, 533, 898, 631, 11

Offset: 1

Clark Kimberling, Jun 24 2011

Row 1:  numbers k such that k = 1 or k = 2 or (k - 1 or k + 1) is a prime.
Row r > 1: numbers k such that k + r or k - r is a prime but k + q and k - q are not, for q = 1, 2, ..., r - 1.
Every positive integer occurs exactly once, so that as a sequence, A192178 is a permutation of the positive integers.
For r > 1, the numbers in row r have the parity of r - 1; e.g., the numbers in row 2 are odd.

			Northwest corner:
1....2....3....4....6....8....10
5....7....9....11...13...15...17
26...34...50...56...64...76...86
23...37...47...67...79...83...93
118..122..144..186..204..206..216
...
For example, 34 is in row 3 recause its distance to the nearest prime is 3.

Cf. A192175, A192176, A192177, A192179.

z = 5000; (* z = number of primes used *)
row[1] = (#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[#1 - 1] || PrimeQ[#1 + 1] || #1 == 1 || #1 == 2} &, {z}], {_, True}];
Do[row[x] = Complement[(#1[[1]] &) /@ Cases[Array[{#1, PrimeQ[#1 - x] || PrimeQ[#1 + x]} &, {z}], {_, True}], Flatten[Array[row, {x - 1}]]], {x, 2, 10}];
TableForm[Array[row, {10}]]  (* A192178 array *)
Flatten[Table[row[k][[n - k + 1]], {n, 1, 11}, {k, 1,
   n}]]   (* A192178 sequence *)
(* by Peter J. C. Moses, Jun 24 2011 *)

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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

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A192175 Array of primes determined by distance to next prime, by antidiagonals.

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A192176 Index array for A192175 (distance up to next prime), by antidiagonals.

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A192177 Array determined by distance down to nearest prime.

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A192178 Array by distance to nearest prime, by antidiagonals.

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