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

Author

M. F. Hasler, Sep 26 2017

Keywords

• nonn

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?

Examples

			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, ...]
		

Links

• 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.

Crossrefs

Cf. A008284, A000041.

Programs

PARI

for(n=1,999,#Set(A008284(n))A008284(n) is the n-th row of A008284.