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.

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A342267 Number of partitions of 1/n into n reciprocals of positive integers.

Original entry on oeis.org

1, 2, 21, 694, 118995, 132891609
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 07 2021

Keywords

Examples

			a(2) = 2 because we have 1/2 = 1/4 + 1/4 = 1/3 + 1/6.

Crossrefs

Cf. A002966, A003167, A004194, A018892, A020327, A020328, A343074, A343214, A369470.

Extensions

a(6) from Jud McCranie, Sep 02 2021

A063715 Number of solutions (r,u,s,t) in positive integers to the system of equations 1/r + 1/u = 1/n, 1/s + 1/t = 1/u.

Original entry on oeis.org

3, 17, 29, 49, 45, 111, 37, 115, 109, 159, 69, 319, 45, 207, 279, 191, 69, 367, 69, 487, 315, 183, 93, 681, 141, 235, 291, 495, 117, 909, 53, 357, 331, 259, 559, 967, 45, 279, 459, 949, 117, 1025, 69, 663, 815, 219, 117, 1205, 161, 591, 411, 555, 93, 965, 579
Offset: 1

Views

Author

Vladeta Jovovic, Aug 10 2001

Keywords

Examples

			For n=2 we have 17 solutions (r,u,s,t): (3,6,7,42), (3,6,8,24), (3,6,9,18), (3,6,10,15), (3,6,12,12), (3,6,15,10), (3,6,18,9), (3,6,24,8), (3,6,42,7), (4,4,5,20), (4,4,6,12), (4,4,8,8), (4,4,12,6), (4,4,20,5), (6,3,4,12), (6,3,6,6), (6,3,12,4).

Crossrefs

Cf. A000005, A048691, A063716, A048691, A004194, A063520.

Programs

  • PARI
    a(n) = sumdiv(n^2, d, numdiv((n+d)^2)) \\ Michel Marcus, Jun 17 2013

Formula

a(n) = Sum_{d|n^2} tau((n+d)^2).

A063716 Number of solutions (r,u,s,t), s>=t, in positive integers to the system of equations 1/r+1/u=1/n, 1/s+1/t=1/u.

Original entry on oeis.org

2, 10, 16, 27, 24, 60, 20, 61, 57, 84, 36, 167, 24, 108, 144, 100, 36, 191, 36, 251, 162, 96, 48, 351, 73, 122, 149, 255, 60, 468, 28, 184, 170, 134, 284, 496, 24, 144, 234, 485, 60, 526, 36, 339, 415, 114, 60, 616, 83, 303, 210, 285, 48, 493, 294, 585, 210, 144, 96
Offset: 1

Views

Author

Vladeta Jovovic, Aug 10 2001

Keywords

Examples

			For n=2 we have 10 solutions (r,u,s,t), s>=t: (3,6,12,12), (3,6,15,10), (3,6,18,9), (3,6,24,8), (3,6,42,7), (4,4,8,8), (4,4,12,6), (4,4,20,5), (6,3,6,6), (6,3,12,4).

Crossrefs

Cf. A000005, A048691, A063715, A018892, A048691, A004194, A063520.

Formula

a(n) = (tau(n^2) + Sum_{d|n^2} tau((n+d)^2))/2 = (A048691(n)+A063715(n))/2.

A082604 a(n) is the total number of positive integral solutions (x,y), order being taken into account, to the equation 1/x + 1/y + 1/z = 1/n.

Original entry on oeis.org

10, 46, 106, 148, 196, 310, 232, 394, 442, 538, 352, 910, 334, 778, 1126, 736, 418, 1210, 460, 1702, 1414, 904, 538, 2128, 904, 1006, 1336, 1990, 616, 2884, 520, 1576, 1876, 1228, 2668, 3142, 490, 1258, 2212, 3556, 706, 3640, 604, 2782, 3808, 1186, 790
Offset: 1

Views

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), May 23 2003

Keywords

Crossrefs

Cf. A082986.

Formula

a(n) = 6*(A004194(n)-A066761(n))-8. - Vladeta Jovovic, May 26 2003

Extensions

More terms from Vladeta Jovovic, May 26 2003
Previous Showing 11-14 of 14 results.

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