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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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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| 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 number of elements, here 2.
Rows are the same as the rows of A115623, but in reverse order.
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. - W. Lang, Mar 17 2011.
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LINKS
| 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.
W. Lang: First 10 rows.
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FORMULA
| 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
| 0; 1; 1,1; 1,2,1; 1,2,1,2,1; 1,2,2,2,2,2,1; ...
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
| Cf. A036036, A000041, A115623, A115621, row sums A000070.
Sequence in context: A003842 A095771 A007421 * A115623 A134265 A182858
Adjacent sequences: A103918 A103919 A103920 * A103922 A103923 A103924
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KEYWORD
| nonn,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Mar 24 2005
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EXTENSIONS
| Edited by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), May 29 2006
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