cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A182944 Square array A(i,j), i >= 1, j >= 1, of prime powers prime(i)^j, by descending antidiagonals.

Original entry on oeis.org

2, 4, 3, 8, 9, 5, 16, 27, 25, 7, 32, 81, 125, 49, 11, 64, 243, 625, 343, 121, 13, 128, 729, 3125, 2401, 1331, 169, 17, 256, 2187, 15625, 16807, 14641, 2197, 289, 19, 512, 6561, 78125, 117649, 161051, 28561, 4913, 361, 23
Offset: 1

Views

Author

Clark Kimberling, Dec 14 2010

Keywords

Comments

We alternatively refer to this sequence as a triangle T(.,.), with T(n,k) = A(k,n-k+1) = prime(k)^(n-k+1).
The monotonic ordering of this sequence, prefixed by 1, is A000961.
The joint-rank array of this sequence is A182869.
Main diagonal gives A062457. - Omar E. Pol, Sep 11 2018

Examples

			Square array A(i,j) begins:
  i \ j: 1      2      3      4      5  ...
  ---\-------------------------------------
  1:     2,     4,     8,    16,    32, ...
  2:     3,     9,    27,    81,   243, ...
  3:     5,    25,   125,   625,  3125, ...
  4:     7,    49,   343,  2401, 16807, ...
  ...
The triangle T(n,k) begins:
  n\k:  1     2     3     4     5     6  ...
  1:    2
  2:    4     3
  3:    8     9     5
  4:   16    27    25     7
  5:   32    81   125    49    11
  6:   64   243   625   343   121    13
  ...

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
  • Michael De Vlieger, Diagram showing row n of triangle in a semicircle as noted, with a color function associated with the magnitude of T(n,k) compared to 2^n in light blue, where prime(n) is the smallest and the prime power indicated in red the largest in the row.

Crossrefs

Cf. A000961, A006939 (row products of triangle), A062457, A182945, A332979 (row maxima of triangle).
Columns: A000040 (1), A001248 (2), A030078 (3), A030514 (4), A050997 (5), A030516 (6), A092759 (7), A179645 (8), A179665 (9), A030629 (10).
A319075 extends the array with 0th powers.
Subtable of A242378, A284457, A329332.

Programs

  • Mathematica
    TableForm[Table[Prime[n]^j,{n,1,14},{j,1,8}]]

Formula

From Peter Munn, Dec 29 2019: (Start)
A(i,j) = A182945(j,i) = A319075(j,i).
A(i,j) = A242378(i-1,2^j) = A329332(2^(i-1),j).
A(i,i) = A062457(i).
(End)

Extensions

Clarified in respect of alternate reading as a triangle by Peter Munn, Aug 28 2022

A182908 Rank of 2^n when all prime powers (A246655) p^n, for n>=1, are jointly ranked.

Original entry on oeis.org

1, 3, 6, 10, 18, 27, 44, 70, 117, 198, 340, 604, 1078, 1961, 3590, 6635, 12370, 23150, 43579, 82267, 155921, 296347, 564688, 1078555, 2064589, 3958999, 7605134, 14632960, 28195586, 54403835, 105102701, 203287169, 393625231, 762951922, 1480223716, 2874422303
Offset: 1

Views

Author

Clark Kimberling, Dec 13 2010

Keywords

Examples

			a(3)=6 because 2^3 has rank 6 in the sequence (2,3,4,5,7,8,9,...).

Links

Crossrefs

Cf. A000720, A024622, A025528, A246655.
Row 1 of A182869. Complement of A182909.

Programs

  • Mathematica
    T[i_,j_]:=Sum[Floor[j*Log[Prime[i]]/Log[Prime[h]]],{h,1,PrimePi[Prime[i]^j]}]; Flatten[Table[T[i,j],{i,1,1},{j,1,22}]]
f[n_] := Sum[ PrimePi[ Floor[2^(n/k)]], {k, n + 1}]; Array[f, 34] (* Robert G. Wilson v, Jul 08 2011 *)
  • Python
    from sympy import primepi, integer_nthroot
def A182908(n):
    x = 1<Chai Wah Wu, Nov 05 2024

Formula

a(n) = A182908(n) = A024622(n) - 1 for n>=1.
a(n) = Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n). - Ridouane Oudra, Oct 26 2020
a(n) = A025528(2^n). - Pontus von Brömssen, Sep 27 2024

Extensions

Minor edits by Ray Chandler, Aug 20 2021

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

Views

Author

Clark Kimberling, Dec 09 2010

Keywords

Comments

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

Examples

			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...

Crossrefs

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

Programs

  • Mathematica
    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}]]

Extensions

Corrected and extended by Clark Kimberling, Dec 14 2010

A182945 Array of prime powers p^j, as transpose of A182944.

Original entry on oeis.org

2, 3, 4, 5, 9, 8, 7, 25, 27, 16, 11, 49, 125, 81, 32, 13, 121, 343, 625, 243, 64, 17, 169, 1331, 2401, 3125, 729, 128, 19, 289, 2197, 14641, 16807, 15625, 2187, 256, 23, 361, 4913, 28561, 161051, 117649, 78125, 6561, 512, 29, 529, 6859, 83521, 371293, 1771561, 823543, 390625, 19683, 1024
Offset: 1

Views

Author

Clark Kimberling, Dec 14 2010

Keywords

Comments

The monotonic ordering of this sequence, with 1 prefixed, is A000961.
The joint-rank array of this sequence is A182869.

Examples

			Northwest corner:
   2    3     5     7
   4    9    25    49
   8   27   125   343
  16   81   625  2401

Links

Crossrefs

Cf. A000961, A182944, A000040 (row 1), A001248 (row 2), A030078 (row 3).
Antidiagonal products give A006939.
Cf. A319075 (extends the array with 0th powers).

Programs

  • Magma
    [NthPrime(n-i)^i: i in [1..n-1], n in [2..15]]; // Vincenzo Librandi, Jul 28 2015
  • Maple
    seq(seq(ithprime(n-i)^i,i=1..n-1),n=2..20); # Robert Israel, Jul 27 2015
  • Mathematica
    width=9;Table[Table[Prime[n]^j,{n,1,width},{j,1,width}]]; Flatten[Table[Table[%[[z-k+1]][[k]],{k,1,z}],{z,1,width}]]

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

Views

Author

Clark Kimberling, Dec 09 2010

Keywords

Comments

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

Examples

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

Crossrefs

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

Views

Author

Clark Kimberling, Dec 09 2010

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A182801, A182869, A182870, A182871.
Showing 1-6 of 6 results.

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