A257053 -id:A257053 - cp's OEIS frontend

A065730 Largest square <= n-th prime.

1, 1, 4, 4, 9, 9, 16, 16, 16, 25, 25, 36, 36, 36, 36, 49, 49, 49, 64, 64, 64, 64, 81, 81, 81, 100, 100, 100, 100, 100, 121, 121, 121, 121, 144, 144, 144, 144, 144, 169, 169, 169, 169, 169, 196, 196, 196, 196, 225, 225, 225, 225, 225, 225, 256, 256, 256, 256, 256

Offset: 1

Labos Elemer, Nov 15 2001

For n > 2: a(n) = A257053(n,0). - Reinhard Zumkeller, Apr 15 2015

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A048760, A000040, A065731-A065741.

Cf. A257053.

a065730 = a048760 . a000040  -- Reinhard Zumkeller, Apr 15 2015

[Floor(Sqrt(NthPrime(n)))^2: n in [1..70]]; // Vincenzo Librandi, Jan 05 2018

Table[Floor[Sqrt@ Prime@ n]^2, {n, 60}] (* Michael De Vlieger, Jul 03 2016 *)

a(n) = { sqrtint(prime(n))^2 } \\ Harry J. Smith, Oct 28 2009

a(n) = A048760(A000040(n)).

A256913 Enhanced squares representations for k = 0, 1, 2, ..., concatenated.

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 1, 4, 2, 4, 3, 4, 3, 1, 9, 9, 1, 9, 2, 9, 3, 9, 4, 9, 4, 1, 9, 4, 2, 16, 16, 1, 16, 2, 16, 3, 16, 4, 16, 4, 1, 16, 4, 2, 16, 4, 3, 16, 4, 3, 1, 25, 25, 1, 25, 2, 25, 3, 25, 4, 25, 4, 1, 25, 4, 2, 25, 4, 3, 25, 4, 3, 1, 25, 9, 25, 9, 1, 36

Offset: 0

Clark Kimberling, Apr 14 2015

Let B = {0,1,2,3,4,9,16,25,...}, so that B consists of the squares together with 2 and 3.  We call B the enhanced basis of squares.  Define R(0) = 0 and R(n) = g(n) + R(n - g(n)) for n > 0, where g(n) is the greatest number in B that is <= n.  Thus, each n has an enhanced squares representation of the form R(n) = b(m(n)) + b(m(n-1)) + ... + b(m(k)), where b(n) > m(n-1) > ... > m(k) > 0, in which the last term, b(m(k)), is the trace.
The least n for which R(n) has 5 terms is given by R(168) = 144 + 16 + 4 + 3 + 1.
The least n for which R(n) has 6 terms is given by R(7224) = 7056 + 144 + 16 + 4 + 3 + 1.

			R(0) = 0
R(1) = 1
R(2) = 2
R(3) = 3
R(4) = 4
R(8) = 4 + 3 + 1
R(24) = 16 + 4 + 3 + 1

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

Cf. A000290, A256914 (trace), A256915 (number of terms), A256789 (minimal alternating squares representations).

Cf. A257053 (primes).

a256913 n k = a256913_tabf !! n !! k
a256913_row n = a256913_tabf !! n
a256913_tabf = [0] : tail esr where
   esr = (map r [0..8]) ++
           f 9 (map fromInteger $ drop 3 a000290_list) where
     f x gs@(g:hs@(h:_))
       | x < h   = (g : genericIndex esr (x - g)) : f (x + 1) gs
       | otherwise = f x hs
     r 0 = []; r 8 = [4, 3, 1]
     r x = q : r (x - q) where q = [0,1,2,3,4,4,4,4,4] !! x
-- 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 *)

A257070 Traces of primes in enhanced squares representation, cf. A256913.

Original entry on oeis.org

2, 3, 1, 3, 2, 4, 1, 3, 3, 4, 2, 1, 1, 3, 2, 4, 1, 3, 3, 3, 9, 2, 2, 1, 16, 1, 3, 3, 9, 4, 2, 1, 16, 2, 1, 3, 4, 3, 3, 4, 1, 3, 2, 1, 1, 3, 2, 2, 2, 4, 1, 1, 16, 1, 1, 3, 4, 2, 1, 25, 2, 4, 2, 2, 1, 3, 3, 4, 3, 25, 4, 1, 2, 3, 2, 2, 3, 36, 1, 9, 3, 1, 2, 1

Offset: 1

Reinhard Zumkeller, Apr 15 2015

nonn

a(n) = A256914(A000040(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257053, A256913, A000040, A256914.

Haskell
```
a257070 = last . a257053_row
```

A257071 Length of enhanced squares representation of n-th prime, cf. A256913.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 3, 2, 3, 2, 4, 2, 4, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 4, 2, 3, 3, 4, 5, 2, 2, 4, 3, 2, 2, 4, 4, 2, 3, 2, 3, 3, 4, 4, 2, 3, 2, 3, 4, 5, 3, 3, 3, 4, 2, 3, 4, 3, 3, 3, 4, 3, 2, 2, 2, 3, 4, 4, 5, 3, 2

Offset: 1

Reinhard Zumkeller, Apr 15 2015

nonn

a(n) = A256915(A000040(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A257053, A256913, A000040, A256915.

Haskell
```
a257071 = length . a257053_row
```

cp's OEIS Frontend

A065730 Largest square <= n-th prime.

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A256913 Enhanced squares representations for k = 0, 1, 2, ..., concatenated.

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A257070 Traces of primes in enhanced squares representation, cf. A256913.

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A257071 Length of enhanced squares representation of n-th prime, cf. A256913.

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