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Number of partitions of n such that all parts of a partition have the same digital root.
13

%I #7 Feb 04 2014 08:57:17

%S 1,2,2,3,2,4,2,4,3,4,3,7,4,5,7,6,5,8,5,8,11,8,7,16,9,10,13,12,10,22,

%T 11,15,23,16,16,26,16,18,32,22,21,41,24,27,40,28,26,55,30,36,59,40,38,

%U 65,41,45,77,48,51,95,57,60,97,66,63,119,68,80,131,89,85,150,91,96,166,104

%N Number of partitions of n such that all parts of a partition have the same digital root.

%C a(n) = A116371(n) + A116372(n) + A116373(n) + A116374(n) + A116375(n) + A116376(n) + A116377(n) + A116378(n) + A114099(n).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>

%e a(10) = #{10, 5+5, 2+2+2+2+2, 10x1} = 4;

%e a(11) = #{11, 10+1, 11x1} = 3;

%e a(12) = #{12, 10+1+1, 6+6, 4+4+4, 3+3+3+3, 2+2+2+2+2+2, 12x1} = 7.

%o (Haskell)

%o a114102 n = length $ filter (== 1) $

%o map (length . nub . (map a010888)) $ ps 1 n

%o where ps x 0 = [[]]

%o ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]

%o -- _Reinhard Zumkeller_, Feb 04 2014

%Y Cf. A010888.

%Y A147706. [From _Reinhard Zumkeller_, Nov 11 2008]

%Y A156144. [From _Reinhard Zumkeller_, Feb 05 2009]

%K nonn,base

%O 1,2

%A _Reinhard Zumkeller_, Feb 12 2006