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A292917 Numbers n for which the n-th row of A008284 (partitions of n into k parts) has duplicate values > 1. 0
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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A201472 A005049 A128427 * A120182 A037361 A202014
KEYWORD
nonn
AUTHOR
M. F. Hasler, Sep 26 2017
STATUS
approved

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Last modified May 24 14:39 EDT 2024. Contains 372778 sequences. (Running on oeis4.)