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 1-5 of 5 results.

A277239 Number A(n,k) of factorizations of m^n into exactly k factors, where m is a product of two distinct primes; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 5, 1, 0, 1, 2, 8, 8, 1, 0, 1, 2, 9, 19, 13, 1, 0, 1, 2, 9, 27, 42, 18, 1, 0, 1, 2, 9, 30, 74, 78, 25, 1, 0, 1, 2, 9, 31, 95, 168, 139, 32, 1, 0, 1, 2, 9, 31, 105, 248, 363, 224, 41, 1, 0, 1, 2, 9, 31, 108, 300, 614, 703, 350, 50, 1, 0
Offset: 0

Views

Author

Alois P. Heinz, Oct 06 2016

Keywords

Examples

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,    1,    1,    1, ...
  0, 1,  2,   2,    2,    2,    2,    2,    2, ...
  0, 1,  5,   8,    9,    9,    9,    9,    9, ...
  0, 1,  8,  19,   27,   30,   31,   31,   31, ...
  0, 1, 13,  42,   74,   95,  105,  108,  109, ...
  0, 1, 18,  78,  168,  248,  300,  325,  335, ...
  0, 1, 25, 139,  363,  614,  814,  938, 1002, ...
  0, 1, 32, 224,  703, 1367, 1996, 2457, 2741, ...
  0, 1, 41, 350, 1297, 2879, 4642, 6128, 7168, ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000012, A000982(n+1), A101427, A277240, A277241, A277242, A277243, A277244, A277245, A277246.
Main diagonal gives A254686.
A(n,2n) gives A002774.

A101423 Number of different cuboids with volume p^3 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

3, 6, 12, 19, 28, 38, 51, 64, 80, 97, 116, 136, 159, 182, 208, 235, 264, 294, 327, 360, 396, 433, 472, 512, 555, 598, 644, 691, 740, 790, 843, 896, 952, 1009, 1068, 1128, 1191, 1254, 1320, 1387, 1456, 1526, 1599, 1672, 1748, 1825, 1904, 1984, 2067, 2150, 2236
Offset: 0

Views

Author

Anthony C Robin, Jan 17 2005

Keywords

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Crossrefs

Cf. A034836, A101424, A101425, A101426, A101427.

Formula

a(n) = A034836(2^3*3^n) = A034836(3^3*2^n) = A034836(p^3*q^n) for p,q distinct primes.
Empirical g.f.: -(x^3+3*x^2+3*x+3)/((x+1)*(x^2+x+1)*(x-1)^3). - Colin Barker, Mar 28 2014

Extensions

Corrected, edited and extended by Ray Chandler, Dec 19 2008
a(0)=3 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101424 Number of different cuboids with volume p^4 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

4, 9, 18, 28, 42, 57, 76, 96, 120, 145, 174, 204, 238, 273, 312, 352, 396, 441, 490, 540, 594, 649, 708, 768, 832, 897, 966, 1036, 1110, 1185, 1264, 1344, 1428, 1513, 1602, 1692, 1786, 1881, 1980, 2080, 2184, 2289, 2398, 2508, 2622, 2737, 2856, 2976, 3100
Offset: 0

Views

Author

Anthony C Robin, Jan 17 2005

Keywords

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Crossrefs

Cf. A034836, A101423, A101425, A101426, A101427.

Formula

a(n) = A034836(2^4*3^n) = A034836(3^4*2^n) = A034836(p^4*q^n) for p,q distinct primes.
From Colin Barker, Mar 28 2014: (Start)
The following is conjectured.
a(n) = (29 + 3*(-1)^n + 36*n + 10*n^2)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -(x+4)/((x+1)*(x-1)^3). (End)

Extensions

Edited and extended by Ray Chandler, Dec 19 2008
a(0)=4 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101425 Number of different cuboids with volume p^5 X q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

5, 12, 24, 38, 57, 78, 104, 132, 165, 200, 240, 282, 329, 378, 432, 488, 549, 612, 680, 750, 825, 902, 984, 1068, 1157, 1248, 1344, 1442, 1545, 1650, 1760, 1872, 1989, 2108, 2232, 2358, 2489, 2622, 2760, 2900, 3045, 3192, 3344, 3498, 3657, 3818, 3984, 4152
Offset: 0

Views

Author

Anthony C Robin, Jan 17 2005

Keywords

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Crossrefs

Cf. A034836, A101423, A101424, A101426, A101427.

Formula

a(n) = A034836(2^5*3^n) = A034836(3^5*2^n) = A034836(p^5*q^n) for p,q distinct primes.
From Colin Barker, Mar 28 2014: (Start)
The following is conjectured.
a(n) = (37+3*(-1)^n+48*n+14*n^2)/8.
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4).
G.f.: -(2*x+5)/((x+1)*(x-1)^3). (End)

Extensions

Edited and extended by Ray Chandler, Dec 19 2008
a(0)=5 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016

A101426 Number of different cuboids with volume p^6 * q^n, where p,q are distinct prime numbers.

Original entry on oeis.org

7, 16, 32, 51, 76, 104, 139, 176, 220, 267, 320, 376, 439, 504, 576, 651, 732, 816, 907, 1000, 1100, 1203, 1312, 1424, 1543, 1664, 1792, 1923, 2060, 2200, 2347, 2496, 2652, 2811, 2976, 3144, 3319, 3496, 3680, 3867, 4060, 4256, 4459, 4664, 4876, 5091, 5312
Offset: 0

Views

Author

Anthony C Robin, Jan 17 2005

Keywords

Comments

Subsequence of A034836, which gives the number of cuboids for volume n.

Crossrefs

Cf. A034836, A101423, A101424, A101425, A101427.

Formula

a(n) = A034836(2^6*3^n) = A034836(3^6*2^n) = A034836(p^6*q^n) for p,q distinct primes.
Empirical g.f.: -(3*x^3+9*x^2+9*x+7)/((x+1)*(x^2+x+1)*(x-1)^3). - Colin Barker, Mar 28 2014

Extensions

Edited and extended by Ray Chandler, Dec 19 2008
a(0)=7 prepended and g.f. edited by Alois P. Heinz, Oct 05 2016
Showing 1-5 of 5 results.

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