A094761 -id:A094761 - cp's OEIS frontend

A094763 Trajectory of 2 under repeated application of the map n -> n + square excess of n.

2, 3, 5, 6, 8, 12, 15, 21, 26, 27, 29, 33, 41, 46, 56, 63, 77, 90, 99, 117, 134, 147, 150, 156, 168, 192, 215, 234, 243, 261, 266, 276, 296, 303, 317, 345, 366, 371, 381, 401, 402, 404, 408, 416, 432, 464, 487, 490, 496, 508, 532, 535, 541, 553, 577, 578, 580, 584, 592, 608

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

Robert Israel, Table of n, a(n) for n = 0..10000
S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094761, A094764.

a[0]:= 2:
for n from 1 to 100 do a[n]:= f(a[n-1]) od:
seq(a[n],n=0..100); # Robert Israel, Jan 28 2018

lista(nn) = {print1(n=2, ", "); for (k=2, nn, m = 2*n - sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 23 2015

A094764 Trajectory of 7 under repeated application of the map n --> n + square excess of n.

Original entry on oeis.org

7, 10, 11, 13, 17, 18, 20, 24, 32, 39, 42, 48, 60, 71, 78, 92, 103, 106, 112, 124, 127, 133, 145, 146, 148, 152, 160, 176, 183, 197, 198, 200, 204, 212, 228, 231, 237, 249, 273, 290, 291, 293, 297, 305, 321, 353, 382, 403, 406, 412, 424, 448, 455, 469, 497, 510, 536, 543

Offset: 0

N. J. A. Sloane, Jun 10 2004

nonn

H. Brocard, Note 2837, L'Intermédiaire des Mathématiciens, 11 (1904), p. 239.

S. H. Weintraub, An interesting recursion, Amer. Math. Monthly, 111 (No. 6, 2004), 528-530.

Cf. A053186, A094761, A094763.

lista(nn) = {print1(n=7, ", "); for (k=2, nn, m = 2*n - sqrtint(n)^2; print1(m, ", "); n = m;);} \\ Michel Marcus, Oct 24 2015

A255315 Lower triangular matrix describing the shape of a half hyperbola in the Dirichlet divisor problem.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Mats Granvik, May 31 2015

The sum of terms of row n is n. Length of row n is n.
From Mats Granvik, Feb 21 2016: (Start)
A006218(n) = (n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,k)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))).
A006218(n) = -((n^2 - ((2*Sum_{kk=1..n} Sum_{k=1..kk} T(n,n-k+1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))).
(End)
From Mats Granvik, May 28 2017: (Start)
A006218(n) = (n^2 - (2*(Sum_{k=1..n} T(n, k)*(n - k + 1)) - n)) + 2*n - round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n))).
A006218(n) = -((n^2 - (2*(Sum_{k=1..n} T(n, n - k + 1)*(n - k + 1)) - n)) - 2*n + round(1 + (1/2)*(-3 + sqrt(n) + sqrt(1 + n)))).
(End)
From Mats Granvik, Sep 07 2017: (Start)
It appears that:
The number of 0's in row n is equal to the number of 2's in row n and their number is given by A000196(n) - 1.
The number of 1's in column k is given by A152948(k+2).
The number of 2's in column k is given by A000096(k-1).
The row index of the last nonzero entry in column k is given by A005563(k).
(End)
From Mats Granvik, Oct 06 2018: (Start)
The smallest k such that T(n,k)=2 is given by A079643(n) = floor(n/floor(sqrt(n))).
This gives the lower bound: A006218(n) >= A094761(n) + A079643(n)*2*(A000196(n)-1).
<=> A006218(n) >= 2*n - (floor(sqrt(n)))^2 + floor(n/floor(sqrt(n)))*2*floor(sqrt(n)-1).
The average of k:s such that T(n,k)=2, for n>3 is given by:
b(n) = Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2)))/floor(sqrt(n)-1).
This gives A006218(n) = 2*n - (floor(sqrt(n)))^2 + b(n)*2*floor(sqrt(n)-1) = 2*n - (floor(sqrt(n)))^2 + (Sum_{k=1..n} (k*floor(abs(T(n, k)-1/2))))*2, for n>3.
The largest k such that T(n,k)=2 is given by A004526(n) = floor(n/2).
This gives the upper bound: A006218(n) <= A094761(n) + A004526(n)*2*(A000196(n)-1).
<=> A006218(n) <= 2*n - (floor(sqrt(n)))^2 + floor(n/2)*2*floor(sqrt(n)-1).
The lower bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 21, 23, ...
Sequence A006218 starts: 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, ...
The upper bound starts:  1, 3, 5, 8, 10, 14, 16, 20, 25, 31, ...
(End)

			1;
1, 1;
1, 1, 1;
0, 2, 1, 1;
0, 2, 1, 1, 1;
0, 1, 2, 1, 1, 1;
0, 1, 2, 1, 1, 1, 1;
0, 1, 1, 2, 1, 1, 1, 1;
0, 0, 2, 2, 1, 1, 1, 1, 1;
0, 0, 2, 1, 2, 1, 1, 1, 1, 1;
0, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1;
0, 0, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1;

Eric Weisstein's World of Mathematics, Dirichlet Divisor Problem

Cf. A000096, A000196, A004526, A005563, A006218, A079643, A094761, A094820, A152948.

(* From Mats Granvik, Feb 21 2016: (Start) *)
nn = 12;
T = Table[
   Sum[Table[
     If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,
       If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,
       1, r}], {k, 1, r}], {r, 1, nn}];
Flatten[T]
A006218a = Table[(n^2 - (2*Sum[Sum[T[[n, k]], {k, 1, kk}], {kk, 1, n}] -
        n)) + 2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n,
     1, nn}];
A006218b = -Table[(n^2 - (2*
          Sum[Sum[T[[n, n - k + 1]], {k, 1, kk}], {kk, 1, n}] - n)) -
     2*n + Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}];
(A006218b - A006218a);
(* (End) *)
(* From Mats Granvik, May 28 2017: (Start) *)
nn = 12;
T = Table[
   Sum[Table[
     If[And[If[n*k <= r, If[n >= k, 1, 0], 0] == 1,
       If[(n + 1)*(k + 1) <= r, If[n >= k, 1, 0], 0] == 0], 1, 0], {n,
       1, r}], {k, 1, r}], {r, 1, nn}];
Flatten[T]
A006218a = Table[(n^2 - (2*Sum[T[[n, k]]*(n - k + 1), {k, 1, n}] - n)) +
    2*n - Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])], {n, 1, nn}];
A006218b = Table[-((n^2 - (2*Sum[T[[n, n - k + 1]]*(n - k + 1), {k, 1, n}] -
          n)) - 2*n +
      Round[1 + (1/2)*(-3 + Sqrt[n] + Sqrt[1 + n])]), {n, 1, nn}];
(A006218b - A006218a);
(* (End) *)

See Mathematica program.

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A094763 Trajectory of 2 under repeated application of the map n -> n + square excess of n.

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A094764 Trajectory of 7 under repeated application of the map n --> n + square excess of n.

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A255315 Lower triangular matrix describing the shape of a half hyperbola in the Dirichlet divisor problem.

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