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%I #23 Jan 02 2023 12:30:54
%S 5,6,7,8,10,11,13,14,15,19,22,23,26,30,31,34,43,44,45,46,60,61,68,84,
%T 85,112,113,154,155,202,203,270,271,352,353,462,463,594,595,770,771,
%U 980,981,1254,1255,1584,1585,2004,2005,2510,2511,3150,3151,3916,3917,4872,4873
%N Numbers n for which the n-th row of A008284 (partitions of n into k parts) has duplicate values > 1.
%C 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.
%C 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).
%C 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?
%C In some rare cases (11, 14, 60) there is more than one pair of repeated values. Are there other such cases beyond 60?
%H M. F. Hasler, in reply to Hans Havermann, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-September/017980.html">Re: Finite Sequence?</a>, Sept. 26, 2017.
%H Jonathan Stauduhar, <a href="https://math.stackexchange.com/questions/2432586/is-this-sequence-of-numbers-related-to-partitions-finite">Is this sequence of numbers related to partitions finite?</a>, Mathematics Stack Exchange, Sept. 17, 2017.
%e Denote by A8284(n) the n-th row of the table A008284. Then, for example:
%e A8284(8) = [1, 4, 5*, 5*, 3, 2, 1, 1]
%e A8284(11) = [1, 5*, 10**, 11, 10**, 7, 5*, 3, 2, 1, 1]
%e A8284(13) = [1, 6, 14*, 18**, 18**, 14*, 11, 7, 5, 3, 2, 1, 1]
%e A8284(14) = [1, 7*, 16, 23**, 23**, 20, 15, 11, 7*, 5, 3, 2, 1, 1]
%e A8284(19) = [1, 9, 30*, 54, 70, 71, 65, 52, 41, 30*, 22, 15, 11, 7, 5, 3, 2, 1, 1]
%e 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.
%e A8284(34) = [1, 17, 96, 297*, 603, 931, 1175, 1297, 1291, 1204, 1060, 905, 747, 608, 483, 383, 297*, 231, 176, 135, 101, ...]
%e 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, ...]
%e 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, ...]
%o (PARI) for(n=1,999,#Set(A008284(n))<n-2 && print1(n",")) \\ where A008284(n) is the n-th row of A008284.
%Y Cf. A008284, A000041.
%K nonn
%O 1,1
%A _M. F. Hasler_, Sep 26 2017
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