A255395 -id:A255395 - cp's OEIS frontend

A255313 Triangle read by rows: row n contains the sums of adjacent pairs of terms in row n of A088643.

3, 5, 3, 7, 5, 3, 7, 5, 7, 5, 11, 7, 5, 7, 5, 13, 11, 7, 5, 7, 5, 13, 11, 13, 11, 7, 5, 3, 17, 13, 11, 13, 11, 7, 5, 3, 19, 17, 13, 11, 13, 11, 7, 5, 3, 19, 17, 19, 17, 13, 11, 7, 5, 7, 5, 23, 19, 17, 19, 17, 13, 11, 7, 5, 7, 5, 23, 19, 17, 19, 23, 19, 13

Offset: 1

Reinhard Zumkeller, Feb 22 2015

All terms are prime by definition of A088643.

See A255313 for sorted distinct terms and A255395 for number of distinct terms.

			.  n |                  T(n,k)                    |       A255316
. ---+--------------------------------------------+----------------------
.  1 |  3                                         | 3
.  2 |  5  3                                      | 3 5
.  3 |  7  5  3                                   | 3 5  7
.  4 |  7  5  7  5                                | 5 7
.  5 | 11  7  5  7  5                             | 5 7 11
.  6 | 13 11  7  5  7  5                          | 5 7 11 13
.  7 | 13 11 13 11  7  5  3                       | 3 5  7 11 13
.  8 | 17 13 11 13 11  7  5  3                    | 3 5  7 11 13 17
.  9 | 19 17 13 11 13 11  7  5  3                 | 3 5  7 11 13 17 19
. 10 | 19 17 19 17 13 11  7  5  7  5              | 5 7 11 13 17 19
. 11 | 23 19 17 19 17 13 11  7  5  7  5           | 5 7 11 13 17 19 23
. 12 | 23 19 17 19 23 19 13 11  7  5  7  5        | 5 7 11 13 17 19 23
. 13 | 23 19 23 19 17 23 19 11  7 11 13  7  3     | 3 7 11 13 17 19 23
. 14 | 29 23 19 23 19 17 23 19 11  7 11 13  7  3  | 3 7 11 13 17 19 23 29

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Cf. A088643, A255316, A255395, A060265.

a255313 n k = a255313_tabl !! (n-1) !! (k-1)
a255313_row n = a255313_tabl !! (n-1)
a255313_tabl = zipWith (zipWith (+)) tss $ map tail tss
               where tss = tail a088643_tabl

(* A is A088643 *)
A[n_, 1] := n;
A[n_, k_] := A[n, k] = For[m = n-1, m >= 1, m--, If[PrimeQ[m + A[n, k-1]] && FreeQ[Table[A[n, j], {j, 1, k-1}], m], Return[m]]];
T[n_] := T[n] = 2 MovingAverage[Table[A[n+1, k], {k, 1, n+1}], {1, 1}];
Array[T, 14] // Flatten (* Jean-François Alcover, Aug 02 2021 *)

T(n,k) = A088643(n,k-1) + A088643(n,k), 1 <= k <= n;

T(n,1) = A060265(n+1);

A255316 Distinct terms in triangle A255313, table read by rows.

Original entry on oeis.org

3, 3, 5, 3, 5, 7, 5, 7, 5, 7, 11, 5, 7, 11, 13, 3, 5, 7, 11, 13, 3, 5, 7, 11, 13, 17, 3, 5, 7, 11, 13, 17, 19, 5, 7, 11, 13, 17, 19, 5, 7, 11, 13, 17, 19, 23, 5, 7, 11, 13, 17, 19, 23, 3, 7, 11, 13, 17, 19, 23, 3, 7, 11, 13, 17, 19, 23, 29, 3, 7, 11, 13, 17

Offset: 1

Reinhard Zumkeller, Feb 22 2015

			See A255313.

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened

Cf. A255313, A255395 (row lengths), A255427 (row products).

import Data.List (nub, sort)
a255316 n k = a255316_tabf !! (n-1) !! (k-1)
a255316_row n = a255316_tabf !! (n-1)
a255316_tabf = map (sort . nub) a255313_tabl

(* A is A088643 *)
A[n_, 1] := n;
A[n_, k_] := A[n, k] = For[m = n - 1, m >= 1, m--,
     If[PrimeQ[m + A[n, k - 1]] &&
     FreeQ[Table[A[n, j], {j, 1, k - 1}], m], Return[m]]];
T[n_] := T[n] = Union[2 MovingAverage[
     Table[A[n+1, k], {k, 1, n+1}], {1, 1}]];
Array[T, 20] // Flatten (* Jean-François Alcover, Aug 02 2021 *)

A255427 a(n) = product of distinct terms of row n in triangle A255313.

Original entry on oeis.org

3, 15, 105, 35, 385, 5005, 15015, 255255, 4849845, 1616615, 37182145, 37182145, 22309287, 646969323, 20056049013, 33426748355, 33426748355, 1236789689135, 1236789689135, 50708377254535, 2180460221945005, 2180460221945005, 102481630431415235

Offset: 1

Reinhard Zumkeller, Feb 22 2015

Product of row n of table A255316;
A001221(a(n)) = A001222(a(n)) = A255395(n);
A006530(a(n)) = A060265(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A255313, A255316, A001221, A001222, A255395.

import Data.List (nub)
a255427 = product . nub . a255313_row

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A255313 Triangle read by rows: row n contains the sums of adjacent pairs of terms in row n of A088643.

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A255316 Distinct terms in triangle A255313, table read by rows.

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A255427 a(n) = product of distinct terms of row n in triangle A255313.

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Haskell