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.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 09 2010

Keywords

Comments

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

Examples

			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...

Crossrefs

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

Programs

  • Mathematica
    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}]]

Formula

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

Extensions

Corrected and extended by Clark Kimberling, Dec 13 2010

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.

A182942 Ranks of primes when all odd prime powers p^j, for j>=1, are jointly ranked.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 87, 88, 89, 90, 91, 92
Offset: 1

Views

Author

Clark Kimberling, Dec 14 2010

Keywords

Comments

Column 1 of the array A182870. Complement of A182943.

Examples

			1,2,3,5,6,7,... are the ranks of 3,5,7,11,13,17...
in 3,5,7,9,11,13,17,...

Crossrefs

Cf. A182870, A182943.

Programs

  • Mathematica
      T[i_,j_]:=Sum[Floor[j*Log[Prime[i+1]]/Log[Prime[h]]],{h,2,PrimePi[Prime[i+1]^j]}]; Table[Flatten[Table[T[i,j],{i,1,80},{j,1,1}]]]

A182943 Ranks of composites when all odd prime powers p^j, for j>=1, are jointly ranked.

Original entry on oeis.org

4, 10, 11, 18, 26, 35, 36, 46, 61, 70, 78, 83, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148
Offset: 1

Views

Author

Clark Kimberling, Dec 14 2010

Keywords

Comments

Complement of A182942.

Crossrefs

Cf. A182870, A182942.

Programs

  • Mathematica
     T[i_,j_]:=Sum[Floor[j*Log[Prime[i+1]]/Log[Prime[h]]],{h,2,PrimePi[Prime[i+1]^j]}]; Complement[Range[200],Table[Flatten[Table[T[i,j],{i,1,80},{j,1,1}]]]]

A182946 Array of odd prime powers p^j, where j>=1, by antidiagonals.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Dec 14 2010

Keywords

Comments

The monotonic ordering of A182946, with 1 prefixed, is A061345. The joint-rank array of A182946 is A182870.

Crossrefs

Cf. A061345, A182445.

Programs

  • Mathematica
     width=9;Transpose[Table[Table[Prime[n+1]^j,{n,1,width},{j,1,width}]]]; Flatten[Table[Table[%[[z-k+1]][[k]],{k,1,z}],{z,1,width}]]
Showing 1-6 of 6 results.

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