A257071 -id:A257071 - cp's OEIS frontend

A257053 Primes in enhanced squares representation, cf. A256913.

2, 3, 4, 1, 4, 3, 9, 2, 9, 4, 16, 1, 16, 3, 16, 4, 3, 25, 4, 25, 4, 2, 36, 1, 36, 4, 1, 36, 4, 3, 36, 9, 2, 49, 4, 49, 9, 1, 49, 9, 3, 64, 3, 64, 4, 3, 64, 9, 64, 9, 4, 2, 81, 2, 81, 4, 3, 1, 81, 16, 100, 1, 100, 3, 100, 4, 3, 100, 9, 100, 9, 4, 121, 4, 2

Offset: 1

Reinhard Zumkeller, Apr 15 2015

A257070(n) = length of n-th row;
T(n,k) = A256913(A000040(n),k), k = 0..A257070(n)-1;
T(n,0) = A065730(n) for n > 2;
T(n,A257071(n)-1) = A257070(n).

			.   n | prime(n) |  ESR, row sum = prime(n)
.  ---+----------+-------------------------
.   1 |        2 |  [2]
.   2 |        3 |  [3]
.   3 |        5 |  [4, 1]
.   4 |        7 |  [4, 3]
.   5 |       11 |  [9, 2]
.   6 |       13 |  [9, 4]
.   7 |       17 |  [16, 1]
.   8 |       19 |  [16, 3]
.   9 |       23 |  [16, 4, 3]
.  10 |       29 |  [25, 4]
.  11 |       31 |  [25, 4, 2]
.  12 |       37 |  [36, 1]
.  13 |       41 |  [36, 4, 1]
.  14 |       43 |  [36, 4, 3]
.  15 |       47 |  [36, 9, 2]
.  16 |       53 |  [49, 4]
.  17 |       59 |  [49, 9, 1]
.  18 |       61 |  [49, 9, 3]
.  19 |       67 |  [64, 3]
.  20 |       71 |  [64, 4, 3]
.  21 |       73 |  [64, 9]
.  22 |       79 |  [64, 9, 4, 2]
.  23 |       83 |  [81, 2]
.  24 |       89 |  [81, 4, 3, 1]
.  25 |       97 |  [81, 16]

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

%Cf. A256913, A000040, A065730, A257070 (traces), A257071 (row lengths).

a257053 n k = a257053_tabf !! (n-1) !! k
a257053_row n = a257053_tabf !! (n-1)
a257053_tabf = map (a256913_row . fromIntegral) a000040_list

A256915 Length of the enhanced squares representation of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 1, 2, 2, 2, 2, 3, 3, 3, 4, 2, 3, 3, 3, 3, 4, 4, 2, 1, 2, 2, 2, 2

Offset: 0

Clark Kimberling, Apr 14 2015

See A256913 for definitions.

			R(0) = 0, so length = 1.
R(1) = 1, so length = 1.
R(8) = 4 + 3 + 1, so length = 3.
R(7224) = 7056 + 144 + 16 + 4 + 3 + 1, so length = 6.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A000290, A256913, A256914 (trace).

Cf. A257071.

a256915 = length . a256913_row  -- Reinhard Zumkeller, Apr 15 2015

b[n_] := n^2; bb = Insert[Table[b[n], {n, 0, 100}]  , 2, 3];
s[n_] := Table[b[n], {k, 1, 2 n + 1}];
h[1] = {0, 1, 2, 3}; h[n_] := Join[h[n - 1], s[n]];
g = h[100]; Take[g, 100]
r[0] = {0}; r[1] = {1}; r[2] = {2}; r[3] = {3}; r[8] = {4, 3, 1};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
t = Table[r[n], {n, 0, 120}] (* A256913, before concatenation *)
Flatten[t]  (* A256913 *)
Table[Last[r[n]], {n, 0, 120}]    (* A256914 *)
Table[Length[r[n]], {n, 0, 200}]  (* A256915 *)

cp's OEIS Frontend

A257053 Primes in enhanced squares representation, cf. A256913.

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