A182801 -id:A182801 - cp's OEIS frontend

A182869 Joint-rank array of prime powers: p(i)^j, i>=1, j>=1, read by antidiagonals.

1, 3, 2, 6, 7, 4, 10, 15, 14, 5, 18, 32, 42, 23, 8, 27, 68, 136, 86, 41, 9, 44, 152, 482, 392, 244, 53, 11, 70, 359, 1880, 2001, 1773, 360, 91, 12, 117, 893, 7771, 11211

Offset: 1

Clark Kimberling, Dec 09 2010

Joint-rank arrays are defined in the first comment at A182801. A182869 is a permutation of the positive integers.

			First, arrange the prime powers in rows:
2....4....8....16....32...
3....9...27....81...243...
5...25..125...625..3125...
Then replace each prime power by its rank when they are all jointly ranked:
1....3....6....10.....18...
2....7...15....32.....68...
4...14...42...136....482...
5...23...86...392...2001...
8...41..244..1773..14901...

Cf. A000961, A027883, A182801, A182870, A182908.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}];
TableForm[Table[T[i,j],{i,1,6},{j,1,6}]]

T(i,j) = Sum_{h>=1} floor(j*log(p(i))/log(p(h))), where p(i) denotes the i-th prime.

Corrected and extended by Clark Kimberling, Dec 13 2010

A182870 Joint-rank array of odd prime powers: p(i+1)^j, i>=1, j>=1, read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 11, 10, 3, 26, 36, 18, 5, 61, 127, 78, 35, 6, 143, 471, 381, 234, 46, 7, 348, 1867, 1987, 1760, 349, 70, 8, 881, 7755, 11195, 14884, 3166, 686, 111, 9, 2279

Offset: 1

Clark Kimberling, Dec 09 2010

Joint-rank arrays are defined in the first comment at A182801. A182870 is a permutation of the positive integers.

			First, arrange the odd prime powers in rows:
3....9...27....81...
5...25..125...625...
7...49..343...2401...
Then replace each by its ranks when they are all jointly ranked:
1....4...11....26...
2...10...36...127...
3...18...78...381...
5...35..234..1760...

A182801, A182869, A182871, A182872, A182942, A182943.

T[i_,j_]:=Sum[Floor[j*Log[Prime[i+1]]/Log[Prime[h]]],{h,2,PrimePi[Prime[i+1]^j]}]; TableForm[Table[T[i,j],{i,1,6},{j,1,6}]]

Corrected and extended by Clark Kimberling, Dec 14 2010

A182846 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(2), i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 7, 4, 9, 13, 11, 6, 12, 19, 21, 17, 8, 16, 26, 32, 30, 23, 10, 20, 35, 44, 46, 39, 29, 14, 24, 42, 55, 61, 59, 50, 36, 15, 28, 51, 67, 77, 81, 75, 62, 41, 18, 33, 60, 82, 95, 102, 100, 90, 72, 49, 22, 38, 69, 93, 113, 125, 128, 120, 106, 84, 56, 25, 43

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801.

			Northwest corner:
1....3....5....9...12...
2....7...13...19...26...
4...11...21...32...44...
6...17...30...46...61...
The numbers j*(i-1+sqrt(2)), approximately:
(for i=1)  1.41, 2.83, 4.24,...
(for i=2)  2.41, 4.83, 7.24,...
(for i=3)  3.41, 6.83, 10.24,...
Replacing each by its rank gives
1....3....5
2....7...13
4...ll...21

Cf. A182801, A182847-A182849.

r=Sqrt[2];
f[i_,j_]:=Sum[Floor[j*(i-1+r)/(k-1+r)],{k,1,1+r+j(i-1+r)}];
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (*A182846*)

T(i,j)=SUM{floor(j*(i-1+r)/(k-1+r)): r=sqrt(2), k>=1} for i>=1, j>=1.

A182849 Joint-rank array of the numbers j*(i-1+r), where r = golden ratio = (1+sqrt(5))/2, and i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 9, 13, 11, 5, 14, 19, 21, 16, 8, 18, 27, 31, 30, 22, 10, 23, 36, 43, 45, 39, 28, 12, 26, 44, 56, 61, 57, 50, 34, 15, 32, 52, 68, 78, 79, 73, 60, 40, 17, 37, 63, 83, 94, 101, 98, 87, 70, 47, 20, 42, 72, 96, 113, 124, 126, 118, 104, 82, 54, 24, 48

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801. Every positive integer occurs exactly once, so that as a sequence, A182849 is a permutation of the positive integers.

			Northwest corner:
1....3....6....9...
2....7...13...19...
4...11...21...31...
5...16...30...45...

Cf. A182801, A182846.

r=GoldenRatio;
f[i_,j_]:=Sum[Floor[j*(i-1+r)/(k-1+r)],{k,1,1+r+j(i-1+r)}];
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (* A182849 *)

T(i,j)=SUM{floor(j*(i-1+r)/(k-1+r)): r=(1+sqrt(5))/2, k>=1} for i>=1, j>=1.

A167267 Interspersion of the signature sequence of (1+sqrt(5))/2.

Original entry on oeis.org

1, 3, 2, 7, 5, 4, 12, 10, 8, 6, 19, 16, 14, 11, 9, 28, 24, 21, 18, 15, 13, 38, 34, 30, 26, 23, 20, 17, 50, 45, 41, 36, 32, 29, 25, 22, 63, 58, 53, 48, 43, 39, 35, 31, 27, 78, 72, 67, 61, 56, 51, 46, 42, 37, 33

Offset: 1

Clark Kimberling, Oct 31 2009

Row n is the ordered sequence of numbers k such that A084531(k)=n.  Is the difference sequence of column 1 equal to A019446? Is the difference sequence of row 1 essentially equal to A026351?
As a sequence, A167267 is a permutation of the positive integers. As an array, A167267 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = golden ratio = (1+sqrt(5))/2. - Clark Kimberling, Nov 10 2012
This is a transposable interspersion; i.e., its transpose, A283734, is also an interspersion. - Clark Kimberling, Mar 16 2017

			Northwest corner:
1....3....7....12...19...28...38
2....5....10...16...24...34...45
4....8....14...21...30...41...53
6....11...18...26...36...48...61
9....15...23...32...43...56...70
13...20...29...39...51...65...80

Clark Kimberling, "Fractal Sequences and Interspersions," Ars Combinatoria 45 (1997) 157-168.

Clark Kimberling, Antidiagonals n = 1..60, flattened
N. Carey, Lambda Words: A Class of Rich Words Defined Over an Infinite Alphabet, J. Int. Seq. 16 (2013) #13.3.4

Cf. A001622, A084531, A084531, A283734.

v = GoldenRatio;
x = Table[Sum[Ceiling[i*v], {i, q}], {q, 0, end = 35}];
y = Table[Sum[Ceiling[i*1/v], {i, q}], {q, 0, end}];
tot[p_, q_] := x[[p + 1]] + p q + 1 + y[[q + 1]]
row[r_] := Table[tot[n, r], {n, 0, (end - 1)/v}]
Grid[Table[row[n], {n, 0, (end - 1)}]]
(* Norman Carey, Jul 03 2012 *)

\\ Produces the triangle when the array is read by antidiagonals
r = (1+sqrt(5))/2;
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i, j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k), ", "); ); print(); ); };
tabl(10) \\ Indranil Ghosh, Mar 26 2017

# Produces the triangle when the array is read by antidiagonals
import math
from sympy import sqrt
r=(1 + sqrt(5))/2
def s(n): return 1 if n<1 else s(n - 1) + 1 + int(math.floor(n*r))
def p(n): return n + 1 + sum(int(math.floor((n - k)/r)) for k in range(n+1))
v=[s(n) for n in range(101)]
u=[p(n) for n in range(101)]
def w(i, j): return u[i - 1] + v[j - 1] + (i - 1) * (j - 1) - 1
for n in range(1, 11):
    print([w(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 26 2017

R(m,n) = sum{[(m-i+n+r)/r], i=1,2,...z(m,n)}, where r = (1+sqrt(5))/2 and z(m,n) = m + [(n-1)*r]. - Clark Kimberling, Nov 10 2012

A182847 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(3), i>=1, j>=1, by diagonals.

Original entry on oeis.org

1, 3, 2, 6, 7, 4, 10, 13, 11, 5, 14, 20, 21, 16, 8, 18, 27, 32, 30, 22, 9, 24, 36, 42, 44, 38, 26, 12, 29, 46, 55, 61, 58, 49, 33, 15, 34, 54, 69, 77, 78, 72, 59, 40, 17, 39, 64, 84, 95, 100, 98, 87, 70, 47, 19, 45, 73, 97, 113, 123, 124, 117, 103, 80

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801.

			Northwest corner:
1....3....6...10...
2....7...13...20...
4...11...21...32...
5...16...30...44...

Cf. A182801, A182846, A182848, A182849.

r=Sqrt[3];
f[i_,j_]:=Sum[Floor[j*(i-1+r)/(k-1+r)],{k,1,1+r+j(i-1+r)}];
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]]

T(i,j)=sum(k>=1, floor( j*(i-1+r)/(k-1+r) ) ) where r=sqrt(3), for i>=1, j>=1.

A182848 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(5), i>=1, j>=1, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 8, 7, 3, 12, 14, 11, 5, 17, 22, 21, 16, 6, 24, 30, 33, 29, 20, 9, 28, 40, 45, 44, 38, 26, 10, 35, 51, 59, 61, 57, 47, 32, 13, 41, 60, 73, 80, 77, 69, 56, 37, 15, 49, 71, 88, 97, 101, 94, 84, 66, 43, 18, 55, 82, 103, 115, 124, 123, 113, 99, 76, 50, 19, 64

Offset: 1

Clark Kimberling, Dec 08 2010

Joint-rank arrays are defined in the first comment at A182801.

			Northwest corner:
1....4....8...12...17...
2....7...14...22...30...
3...11...21...33...45...
5...16...29...44...61...

Cf. A182801, A182846.

r=Sqrt[5];
f[i_,j_]:=Sum[Floor[j*(i-1+r)/(k-1+r)],{k,1,1+r+j(i-1+r)}];
TableForm[Table[f[i,j],{i,1,10},{j,1,10}]]

T(i,j) = Sum_{k>=1} floor(j*(i-1+r)/(k-1+r)), where r=sqrt(5).

A182871 Joint-rank array of the numbers p^j, where p is a prime congruent to 3 mod 4 and j>=1, read by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 11, 4, 16, 27, 20, 5, 26, 36, 32, 28, 6, 30, 49, 47, 42, 29, 8, 34, 59, 61, 60, 46, 31, 9, 41, 70, 75, 78, 64, 55, 33, 10, 52, 85, 89, 96, 86, 71, 57, 35, 12, 56, 94, 103, 114, 102, 92, 80, 58, 37, 13, 62, 106, 119, 129, 121, 113, 101, 84, 63, 38

Offset: 1

Clark Kimberling, Dec 09 2010

Joint-rank arrays are defined in the first comment at A182801. A182871 is a permutation of the positive integers.

			Northwest corner:
1....3....7...16...
2...11...27...36...
4...20...32...47...
5...28...42...60...

Cf. A182801, A182869, A182870, A182872.

A182872 Joint-rank array of the numbers p^j, where p is a prime congruent to 1 mod 4 and j>=1, read by antidiagonals.

Original entry on oeis.org

1, 4, 2, 16, 20, 3, 25, 29, 24, 5, 31, 45, 33, 26, 6, 40, 55, 52, 43, 27, 7, 51, 71, 63, 59, 49, 28, 8, 57, 83, 79, 78, 65, 50, 30, 9, 66, 97, 92, 95, 84, 68, 53, 32, 10, 76, 111, 108, 113, 104, 87, 75, 54, 34, 11, 81, 123, 122, 131, 120, 105, 93, 77, 56

Offset: 1

Clark Kimberling, Dec 09 2010

Joint-rank arrays are defined in the first comment at A182801. A182872 is a permutation of the positive integers.

			Northwest corner:
1....4...16...25...
2...20...29...45...
3...24...33...52...
5...26...43...59...

Cf. A182801, A182869, A182870, A182871.

A283938 Interspersion of the signature sequence of tau^2, where tau = (1 + sqrt(5))/2 = golden ratio.

Original entry on oeis.org

1, 4, 2, 10, 6, 3, 18, 13, 8, 5, 29, 22, 16, 11, 7, 43, 34, 26, 20, 14, 9, 59, 49, 39, 31, 24, 17, 12, 78, 66, 55, 45, 36, 28, 21, 15, 99, 86, 73, 62, 51, 41, 33, 25, 19, 123, 108, 94, 81, 69, 57, 47, 38, 30, 23, 150, 133, 117, 103, 89, 76, 64, 53, 44, 35

Offset: 1

Clark Kimberling, Mar 18 2017

Row n is the ordered sequence of numbers k such that A118276(k) = n. As a sequence, A283938 is a permutation of the positive integers. As an array, A283938 is the joint-rank array (defined at A182801) of the numbers {i+j*r}, for i>=1, j>=1, where r = tau^2 = (3 + sqrt(5))/2. This is a transposable interspersion; i.e., every row intersperses all other rows, and every column intersperses all other columns.

			Northwest corner:
1   4  10   18  29  43  59   78  99   123
2   6  13   22  34  49  66   86  108  133
3   8  16   26  39  55  73   94  117  143
5  11  20   31  45  62  81  103  127  154
7  14  24   36  51  69  89  112  137  165
9  17  28   41  57  76  97  121  147  176
From _Indranil Ghosh_, Mar 19 2017: (Start)
Triangle formed when the array is read by antidiagonals:
    1;
    4,   2;
   10,   6,  3;
   18,  13,  8,  5;
   29,  22, 16, 11,  7;
   43,  34, 26, 20, 14,  9;
   59,  49, 39, 31, 24, 17, 12;
   78,  66, 55, 45, 36, 28, 21, 15;
   99,  86, 73, 62, 51, 41, 33, 25, 19;
  123, 108, 94, 81, 69, 57, 47, 38, 30, 23;
  ...
(End)

Clark Kimberling, Antidiagonals n = 1..60, flattened
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs., Vol. 7, 2004.

Cf. A118276, A283961, A283968, A283969.

r = GoldenRatio^2; z = 100;
s[0] = 1; s[n_] := s[n] = s[n - 1] + 1 + Floor[n*r];
u = Table[n + 1 + Sum[Floor[(n - k)/r], {k, 0, n}], {n, 0, z}] (* A283968, row 1 of A283938 *)
v = Table[s[n], {n, 0, z}] (* A283969, col 1 of A283938 *)
w[i_, j_] := v[[i]] + u[[j]] + (i - 1)*(j - 1) - 1;
Grid[Table[w[i, j], {i, 1, 10}, {j, 1, 10}]] (* A283938, array *)
Flatten[Table[w[k, n - k + 1], {n, 1, 20}, {k, 1, n}]] (* A283938, sequence *)

\\ This code produces the triangle mentioned in the example section
r = (3 +sqrt(5))/2;
z = 100;
s(n) = if(n<1, 1, s(n - 1) + 1 + floor(n*r));
p(n) = n + 1 + sum(k=0, n, floor((n - k)/r));
u = v = vector(z + 1);
for(n=1, 101, (v[n] = s(n - 1)));
for(n=1, 101, (u[n] = p(n - 1)));
w(i,j) = v[i] + u[j] + (i - 1) * (j - 1) - 1;
tabl(nn) = {for(n=1, nn, for(k=1, n, print1(w(n - k + 1, k),", ");); print(););};
tabl(10) \\ Indranil Ghosh, Mar 19 2017

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A182869 Joint-rank array of prime powers: p(i)^j, i>=1, j>=1, read by antidiagonals.

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A182870 Joint-rank array of odd prime powers: p(i+1)^j, i>=1, j>=1, read by antidiagonals.

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A182846 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(2), i>=1, j>=1, by antidiagonals.

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A182849 Joint-rank array of the numbers j*(i-1+r), where r = golden ratio = (1+sqrt(5))/2, and i>=1, j>=1, by antidiagonals.

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A167267 Interspersion of the signature sequence of (1+sqrt(5))/2.

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A182847 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(3), i>=1, j>=1, by diagonals.

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A182848 Joint-rank array of the numbers j*(i-1+r), where r=sqrt(5), i>=1, j>=1, by antidiagonals.

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A182871 Joint-rank array of the numbers p^j, where p is a prime congruent to 3 mod 4 and j>=1, read by antidiagonals.

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