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A103921
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Table of number of distinct parts of partitions of n in Abramowitz-Stegun order.
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9
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0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 3, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 3
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OFFSET
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0,6
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COMMENTS
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The row length sequence of this table is p(n)=A000041(n) (number of partitions).
In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.
Rows are the same as the rows of A115623, but in reverse order.
From Wolfdieter Lang, Mar 17 2011: (Start)
The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).
See a comment by Giovanni Resta under A000005. (End)
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LINKS
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Table of n, a(n) for n=0..104.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 10 rows.
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FORMULA
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a(n, m) = number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).
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EXAMPLE
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Triangle starts:
0,
1,
1, 1,
1, 2, 1,
1, 2, 1, 2, 1,
1, 2, 2, 2, 2, 2, 1,
1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
1, 2, 2, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.
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CROSSREFS
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Cf. A036036, A000041, A115623, A115621, row sums A000070.
Sequence in context: A309414 A007421 A239228 * A115623 A279044 A134265
Adjacent sequences: A103918 A103919 A103920 * A103922 A103923 A103924
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KEYWORD
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nonn,tabf
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AUTHOR
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Wolfdieter Lang, Mar 24 2005
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EXTENSIONS
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Edited by Franklin T. Adams-Watters, May 29 2006
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STATUS
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approved
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