A003603 -id:A003603 - cp's OEIS frontend

A191439 Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, by antidiagonals.

1, 2, 3, 5, 7, 4, 12, 17, 10, 6, 29, 41, 24, 14, 8, 70, 99, 58, 34, 19, 9, 169, 239, 140, 82, 46, 22, 11, 408, 577, 338, 198, 111, 53, 27, 13, 985, 1393, 816, 478, 268, 128, 65, 31, 15, 2378, 3363, 1970, 1154, 647, 309, 157, 75, 36, 16, 5741, 8119, 4756

Offset: 1

Clark Kimberling, Jun 04 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1....2....5....12...29
  3....7....17...41...99
  4....10...24...58...140
  6....14...34...82...198
  8....19...46...111..268

Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.

Cf. A114537, A035513, A035506, A191438.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqr[2];
f[n_] := Floor[n*x+n+1/2] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191439 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191439 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 13, 11, 7, 17, 29, 24, 15, 9, 38, 64, 53, 33, 20, 10, 84, 143, 118, 73, 44, 22, 12, 187, 319, 263, 163, 98, 49, 26, 14, 418, 713, 588, 364, 219, 109, 58, 31, 16, 934, 1594, 1314, 813, 489, 243, 129, 69, 35, 18, 2088, 3564, 2938, 1817

Offset: 1

Clark Kimberling, Jun 05 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...2....4....8...17
  3...6....13...29..64
  5...11...24...53..118
  7...15...33...73..163
  9...20...44...98..219

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12;  x = Sqrt[5];
f[n_] := Floor[n*x] (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191446 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191446 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

Corrected typo in name and fixed Mathematica program by Vaclav Kotesovec, Oct 24 2014

A191540 Dispersion of (floor(2nsqrt(2))), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 14, 22, 11, 6, 39, 62, 31, 16, 7, 110, 175, 87, 45, 19, 9, 311, 494, 246, 127, 53, 25, 10, 879, 1397, 695, 359, 149, 70, 28, 12, 2486, 3951, 1965, 1015, 421, 197, 79, 33, 13, 7031, 11175, 5557, 2870, 1190, 557, 223, 93, 36, 15, 19886, 31607

Offset: 1

Clark Kimberling, Jun 06 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455 and A191536-A191545.

			Northwest corner:
  1,  2,  5,  14,  39, ...
  3,  8, 22,  62, 175, ...
  4, 11, 31,  87, 246, ...
  6, 16, 45, 127, 359, ...
  7, 19, 53, 149, 421, ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of the increasing sequence f[n] *)
r=40; r1=12; c=40; c1=12; f[n_] :=Floor[2n*Sqrt[2]]   (* complement of column 1 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, r1}, {j, 1, c1}]]  (* A191540 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191540 sequence *)

A305820 Filter sequence for a(Fibonacci numbers > 1) = constant sequences.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 5, 2, 6, 7, 8, 9, 2, 10, 11, 12, 13, 14, 15, 16, 2, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 2, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 2, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 2, 82, 83, 84, 85, 86

Offset: 0

Antti Karttunen, Jun 16 2018

nonn

For all i, j:

a(i) = a(j) => A003603(i) = A003603(j) => A304101(i) = A304101(j) => A007895(i) = A007895(j).

Cf. A000045, A003603, A007895, A010056, A072649, A304101.

A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ From A010056
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A305820(n) = if(n<=1, n, if(1==A010056(n),2,2+n-A072649(n)));

a(0)= 0, a(1) = 1, for n > 1, a(n) = 2 if n is a Fibonacci number > 1, otherwise a(n) = 2+n-A072649(n) = running count from 3 onward for non-Fibonacci numbers.

A352537 Primes whose position in the Wythoff array is immediately followed by a prime both in the next column and the next row.

Original entry on oeis.org

2, 3, 919, 1223, 1699, 3329, 8009, 11717, 13691, 19079, 20921, 21011, 22643, 22739, 24623, 26309, 28571, 28619, 28979, 30389, 33629, 34739, 35257, 41179, 42577, 48647, 54133, 58601, 59627, 61511, 65171, 70979, 75707, 80141, 84221, 86869, 90677, 93557, 94781

Offset: 1

Michel Marcus, Mar 20 2022

nonn

			The Wythoff array begins:
   1    2    3    5   ...
   4    7   11   18   ...
   6   10   16   26   ...
   ...
where one can see these 2 patterns:
   2    3   and   3    5
   7             11
so 2 and 3 are terms.

Cf. A003603, A035612, A035513 (Wythoff array).

Intersection of A352538 and A352539.

T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break);););
isokp(m) = my(pos = cell(prime(m))); isprime (T(pos[1], pos[2]+1)) && isprime(T(pos[1]+1, pos[2]));
lista(nn) = for (n=1, nn, if (isokp(n), print1(prime(n), ", ")));

A352539 Primes whose position in the Wythoff array is immediately followed by another prime in the next row.

Original entry on oeis.org

2, 3, 13, 17, 59, 71, 101, 157, 347, 359, 401, 683, 821, 881, 919, 1063, 1223, 1613, 1699, 1787, 1931, 2081, 2333, 2663, 2711, 2909, 2999, 3011, 3299, 3329, 3371, 3389, 3623, 3821, 3911, 4019, 4049, 4337, 4349, 4481, 4931, 5171, 5273, 5651, 5741, 5849, 5879, 6029, 6079

Offset: 1

Michel Marcus, Mar 20 2022

nonn

			The Wythoff array begins:
   1    2    3    5    8   13  ...
   4    7   11   18   29   47  ...
   6   10   16   26   42   68  ...
   ...
So 2, 3 and 13 are terms since they are vertically followed by 7, 11 and 47.

Cf. A003603, A035612, A035513 (Wythoff array).

Cf. A352537 (next row and column), A352538 (next column).

T(n,k) = (n+sqrtint(5*n^2))\2*fibonacci(k+1) + (n-1)*fibonacci(k); \\ A035513
cell(n) = for (r=1, oo, for (c=1, oo, if (T(r,c) == n, return([r, c])); if (T(r,c) > n, break);););
isokv(m) = my(pos = cell(prime(m))); isprime (T(pos[1]+1, pos[2]));
lista(nn) = for (n=1, nn, if (isokv(n), print1(prime(n), ", ")));

A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 7, 9, 8, 19, 12, 16, 14, 10, 32, 20, 27, 24, 17, 13, 53, 33, 45, 40, 29, 22, 15, 87, 54, 74, 66, 48, 37, 25, 18, 142, 88, 121, 108, 79, 61, 41, 30, 21, 231, 143, 197, 176, 129, 100, 67, 50, 35, 23, 375, 232, 320, 286, 210, 163, 109, 82, 58, 38, 26, 608, 376, 519, 464, 341, 265, 177, 134, 95, 62, 43, 28

Offset: 1

Clark Kimberling, Jun 02 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3...6...11..19
  2...4...7...12..20
  5...9...16..27..45
  8...14..24..40..66
  10..17..29..48..79

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=#rows of T, r1=#rows to show *)
c = 40; c1 = 12; (* c=#cols of T, c1=#cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + 3/2]
mex[list_] :=
NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
  Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191427 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191427 sequence *)
(* Peter J. C. Moses, Jun 01 2011 *)

A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 4, 5, 11, 8, 9, 7, 19, 14, 16, 12, 10, 32, 24, 27, 21, 17, 13, 53, 40, 45, 35, 29, 22, 15, 87, 66, 74, 58, 48, 37, 25, 18, 142, 108, 121, 95, 79, 61, 42, 30, 20, 231, 176, 197, 155, 129, 100, 69, 50, 33, 23, 375, 286, 320, 252, 210, 163, 113, 82, 55, 38, 26, 608, 464, 519, 409, 341, 265, 184, 134, 90, 63, 43, 28

Offset: 1

Clark Kimberling, Jun 03 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...3...6...11..19
  2...4...8...14..24
  5...9...16..27..45
  7...12..21..35..58
  10..17..29..48..79

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12;  (* c=# cols of T, c1=# cols to show *)
x = GoldenRatio; f[n_] := Floor[n*x + x]
(* f(n) is complement of column 1 *)
mex[list_] :=
NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
  Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191428 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191428 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 17, 18, 21, 22, 19, 16, 25, 26, 31, 32, 28, 24, 20, 36, 38, 45, 46, 41, 35, 29, 23, 52, 55, 65, 66, 59, 50, 42, 34, 27, 75, 79, 93, 94, 84, 72, 60, 49, 39, 30, 107, 113, 133, 134, 120, 103, 86, 70, 56, 43, 33, 152, 161, 189, 191, 171, 147, 123, 100, 80, 62, 48, 37

Offset: 1

Clark Kimberling, Jun 03 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1...2...4...7...11
  3...5...12..18..18
  6...9...14..21..31
  10..15..22..32..46
  13..19..28..41..59

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12;  (* r=# rows of T to compute, r1=# rows to show *)
c = 40; c1 = 12;   (* c=# cols to compute, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + 3/2] (* f(n) is complement of column 1 *)
mex[list_] :=
NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
  Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];  (* the array T *)
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191430 array *)
Flatten[Table[
  t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191430 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

A191431 Dispersion of ([n*x+x]), where x=sqrt(2) and [ ]=floor, by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 13, 16, 18, 21, 22, 19, 17, 24, 26, 31, 32, 28, 25, 20, 35, 38, 45, 46, 41, 36, 29, 23, 50, 55, 65, 66, 59, 52, 42, 33, 27, 72, 79, 93, 94, 84, 74, 60, 48, 39, 30, 103, 113, 132, 134, 120, 106, 86, 69, 56, 43, 34, 147, 161, 188, 190, 171, 151, 123, 98, 80, 62, 49, 37

Offset: 1

Clark Kimberling, Jun 03 2011

Background discussion:  Suppose that s is an increasing sequence of positive integers, that the complement t of s is infinite, and that t(1)=1.  The dispersion of s is the array D whose n-th row is (t(n), s(t(n)), s(s(t(n))), s(s(s(t(n)))), ...).  Every positive integer occurs exactly once in D, so that, as a sequence, D is a permutation of the positive integers.  The sequence u given by u(n)=(number of the row of D that contains n) is a fractal sequence.  Examples:
(1) s=A000040 (the primes), D=A114537, u=A114538.
(2) s=A022343 (without initial 0), D=A035513 (Wythoff array), u=A003603.
(3) s=A007067, D=A035506 (Stolarsky array), u=A133299.
More recent examples of dispersions: A191426-A191455.

			Northwest corner:
  1.....2....4....7...11...16
  3.....5....8...12...18...26
  6.....9...14...21...31...45
  10...15...22...32...46...66
  13...19...28...41...59...84

Cf. A114537, A035513, A035506.

(* Program generates the dispersion array T of increasing sequence f[n] *)
r = 40; r1 = 12; (* r=# rows of T, r1=# rows to show *)
c = 40; c1 = 12; (* c=# cols of T, c1=# cols to show *)
x = Sqrt[2];
f[n_] := Floor[n*x + x] (* f(n) is complement of column 1 *)
mex[list_] :=
NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1,
  Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[
Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191431 array *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]]
(* A191431 sequence *)
(* Program by Peter J. C. Moses, Jun 01 2011 *)

cp's OEIS Frontend

A191439 Dispersion of ([n*sqrt(2)+n+1/2]), where [ ]=floor, by antidiagonals.

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Mathematica

A191446 Dispersion of [n*sqrt(5)], where [ ]=floor, by antidiagonals.

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Mathematica

Extensions

A191540 Dispersion of (floor(2*n*sqrt(2))), by antidiagonals.

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Mathematica

A305820 Filter sequence for a(Fibonacci numbers > 1) = constant sequences.

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PARI

Formula

A352537 Primes whose position in the Wythoff array is immediately followed by a prime both in the next column and the next row.

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PARI

A352539 Primes whose position in the Wythoff array is immediately followed by another prime in the next row.

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PARI

A191427 Dispersion of ([n*r+3/2]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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Mathematica

A191428 Dispersion of ([n*r+r]), where r=(golden ratio)=(1+sqrt(5))/2 and [ ]=floor, by antidiagonals.

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Mathematica

A191430 Dispersion of ([n*sqrt(2)+3/2]), where [ ]=floor, by antidiagonals.

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A191540 Dispersion of (floor(2nsqrt(2))), by antidiagonals.