Search: a256914 -id:a256914
|
| |
|
|
A256913
|
|
Enhanced squares representations for k = 0, 1, 2, ..., concatenated.
|
|
+10
9
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
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.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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
|
|
|
MATHEMATICA
|
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 *)
Table[Last[r[n]], {n, 0, 120}] (* A256914 *)
Table[Length[r[n]], {n, 0, 200}] (* A256915 *)
|
|
|
PROG
|
(Haskell)
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
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy,tabf,nice
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A256915
|
|
Length of the enhanced squares representation of n.
|
|
+10
4
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,6
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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.
|
|
|
MATHEMATICA
|
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 *)
Table[Last[r[n]], {n, 0, 120}] (* A256914 *)
Table[Length[r[n]], {n, 0, 200}] (* A256915 *)
|
|
|
PROG
|
(Haskell)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A257047
|
|
Numbers not having trace 1 in their enhanced squares representation, see A256913.
|
|
+10
4
|
|
|
|
0, 2, 3, 4, 6, 7, 9, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 25, 27, 28, 29, 31, 32, 34, 36, 38, 39, 40, 42, 43, 45, 47, 48, 49, 51, 52, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 90, 92, 93, 94, 96
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
PROG
|
(Haskell)
a257047 n = a257047_list !! (n-1)
a257047_list = filter ((/= 1) . a256914) [0..]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A257046
|
|
Numbers having trace 1 in their enhanced squares representation, see A256913.
|
|
+10
3
|
|
|
|
1, 5, 8, 10, 14, 17, 21, 24, 26, 30, 33, 35, 37, 41, 44, 46, 50, 54, 57, 59, 63, 65, 69, 72, 74, 78, 82, 86, 89, 91, 95, 98, 101, 105, 108, 110, 114, 117, 122, 126, 129, 131, 135, 138, 142, 145, 149, 152, 154, 158, 161, 165, 168, 170, 174, 177, 179, 183, 186
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
PROG
|
(Haskell)
a257046 n = a257046_list !! (n-1)
a257046_list = filter ((== 1) . a256914) [0..]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A257070
|
|
Traces of primes in enhanced squares representation, cf. A256913.
|
|
+10
3
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
PROG
|
(Haskell)
a257070 = last . a257053_row
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.007 seconds
|
