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 A167932 Number of partitions of n such that all parts are equal or all parts are distinct. 3
 1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 13, 13, 20, 19, 25, 30, 36, 39, 51, 55, 69, 79, 92, 105, 129, 144, 168, 195, 227, 257, 303, 341, 395, 451, 515, 588, 676, 761, 867, 985, 1120, 1261, 1433, 1611, 1821, 2053, 2307, 2591, 2919, 3266, 3663, 4100, 4587, 5121, 5725, 6381 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)). LINKS Table of n, a(n) for n=0..55. Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view) Omar E. Pol, Illustration of the shell model of partitions (2D view) Omar E. Pol, Illustration of the shell model of partitions (3D view) FORMULA a(n) = A000041(n) - A167930(n). a(n) = A000009(n) + A032741(n). EXAMPLE The partitions of 6 are: 6 .............. All parts are distinct ..... (1). 5+1 ............ All parts are distinct ..... (2). 4+2 ............ All parts are distinct ..... (3). 4+1+1 .......... Only some parts are equal. 3+3 ............ All parts are equal ........ (4). 3+2+1 .......... All parts are distinct ..... (5). 3+1+1+1 ........ Only some parts are equal. 2+2+2 .......... All parts are equal ........ (6). 2+2+1+1 ........ Only some parts are equal. 2+1+1+1+1 ...... Only some parts are equal. 1+1+1+1+1+1 .... All parts are equal ........ (7). So a(6) = 7. MATHEMATICA ds[n_]:=Module[{lun=Length[Union[n]]}, Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n], _?(ds)], {n, 0, 60}] (* Harvey P. Dale, Sep 13 2011 *) CROSSREFS Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933. Sequence in context: A325588 A244903 A342337 * A006087 A364675 A241315 Adjacent sequences: A167929 A167930 A167931 * A167933 A167934 A167935 KEYWORD nonn AUTHOR Omar E. Pol, Nov 15 2009 EXTENSIONS More terms from D. S. McNeil, May 10 2010 STATUS approved

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Last modified September 11 05:49 EDT 2024. Contains 375814 sequences. (Running on oeis4.)