A292917 Numbers n for which the n-th row of A008284 (partitions of n into k parts) has duplicate values > 1.

5, 6, 7, 8, 10, 11, 13, 14, 15, 19, 22, 23, 26, 30, 31, 34, 43, 44, 45, 46, 60, 61, 68, 84, 85, 112, 113, 154, 155, 202, 203, 270, 271, 352, 353, 462, 463, 594, 595, 770, 771, 980, 981, 1254, 1255, 1584, 1585, 2004, 2005, 2510, 2511, 3150, 3151, 3916, 3917, 4872, 4873

Offset

1,1

Comments

Let us denote P(n) = A000041(n) the partition numbers, and T(n,k) = A008284(n,k) the number of partitions of n with k parts.

All n = 2*P(k) > 4 (n = 6, 10, 14, 22, 30, 44, 60, 84, 112, 154, 202, ...) and also all n = 2*P(k) + 1 > 4 (n = 5, 7, 11, ...) are in this sequence: In this case, T(n,2) = P(k) = T(n,n-k), cf. formulas for A008284. For example, for n = 2*P(4) = 10, T(10, 2) = 5 = T(10, 6); for n = 2*P(3) + 1 = 7, T(7,2) = 3 = T(7,4).

Some terms (8, 13, 19, 26, 34, 43, 46, 68) are not of the form 2*P(k) or 2*P(k)+1. No such term is known beyond 68: Are there any others?

In some rare cases (11, 14, 60) there is more than one pair of repeated values. Are there other such cases beyond 60?

Links

Table of n, a(n) for n=1..57.

M. F. Hasler, in reply to Hans Havermann, Re: Finite Sequence?, Sept. 26, 2017.

Jonathan Stauduhar, Is this sequence of numbers related to partitions finite?, Mathematics Stack Exchange, Sept. 17, 2017.

Example

Denote by A8284(n) the n-th row of the table A008284. Then, for example:
A8284(8) = [1, 4, 5*, 5*, 3, 2, 1, 1]
A8284(11) = [1, 5*, 10**, 11, 10**, 7, 5*, 3, 2, 1, 1]
A8284(13) = [1, 6, 14*, 18**, 18**, 14*, 11, 7, 5, 3, 2, 1, 1]
A8284(14) = [1, 7*, 16, 23**, 23**, 20, 15, 11, 7*, 5, 3, 2, 1, 1]
A8284(19) = [1, 9, 30*, 54, 70, 71, 65, 52, 41, 30*, 22, 15, 11, 7, 5, 3, 2, 1, 1]
A8284(26) = [1, 13, 56*, 136, 221, 282, 300, 288, 252, 212, 169, 133, 101, 77, 56*, 42, 30, ...], where "..." represents the tail of the preceding list.
A8284(34) = [1, 17, 96, 297*, 603, 931, 1175, 1297, 1291, 1204, 1060, 905, 747, 608, 483, 383, 297*, 231, 176, 135, 101, ...]
A8284(43) = [1, 21, 154, 588, 1469, 2702, 4011, 5066, 5708*, 5888, 5708*, 5262, 4691, 4057, 3446, 2871, 2369, 1928, 1563, 1251, 1001, 792, 627, 490, 385, 297, ...]
A8284(46) = [1, 23, 176*, 720, 1898, 3692, 5731, 7564, 8824, 9418, 9373, 8877, 8073, 7139, 6158, 5231, 4370, 3621, 2965, 2417, 1951, 1573, 1255, 1002, 792, ...]

Prog

(PARI) for(n=1, 999, #Set(A008284(n))<n-2 && print1(n", ")) \\ where A008284(n) is the n-th row of A008284.

Crossrefs

Cf. A008284, A000041.

Sequence in context: A201472 A005049 A128427 * A120182 A037361 A202014

Adjacent sequences: A292914 A292915 A292916 * A292918 A292919 A292920

Keyword

nonn

Author

M. F. Hasler, Sep 26 2017

Status

approved