A116372 -id:A116372 - cp's OEIS frontend

A114099 Number of partitions of n into parts with digital root = 9.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 0, 0, 0, 56, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n).

			a(27) = #{27, 18+9, 9+9+9} = 3.

Antti Karttunen, Table of n, a(n) for n = 0..19683
Eric Weisstein's World of Mathematics, Digital Root

Cf. A000041, A010888, A035444, A147706.

f[n_] := PartitionsP[n/9] If[Mod[n, 9] == 0, 1, 0]; Array[f, 105] (* Robert G. Wilson v, Apr 25 2010 *)

A114099(n) = if((n%9), 0, numbpart(n/9)); \\ Antti Karttunen, Jul 22 2018

a(n) = A000041(floor(n/9))*0^(n mod 9).

a(9n) = A000041(n) and for all others a(n) = 0. [Robert G. Wilson v, Apr 25 2010]

A114102 Number of partitions of n such that all parts of a partition have the same digital root.

Original entry on oeis.org

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, 11, 15, 23, 16, 16, 26, 16, 18, 32, 22, 21, 41, 24, 27, 40, 28, 26, 55, 30, 36, 59, 40, 38, 65, 41, 45, 77, 48, 51, 95, 57, 60, 97, 66, 63, 119, 68, 80, 131, 89, 85, 150, 91, 96, 166, 104

Offset: 1

Reinhard Zumkeller, Feb 12 2006

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

			a(10) = #{10, 5+5, 2+2+2+2+2, 10x1} = 4;
a(11) = #{11, 10+1, 11x1} = 3;
a(12) = #{12, 10+1+1, 6+6, 4+4+4, 3+3+3+3, 2+2+2+2+2+2, 12x1} = 7.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
A156144. [From Reinhard Zumkeller, Feb 05 2009]

a114102 n = length $ filter (== 1) $
            map (length . nub . (map a010888)) $ ps 1 n
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Feb 04 2014

A116371 Number of partitions of n into parts with digital root = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 12, 12, 12, 13, 15, 17, 18, 19, 19, 19, 19, 19, 20, 23, 26, 28, 29, 30, 30, 30, 30, 31, 34, 38, 41, 43, 44, 45, 45, 45, 46, 50, 55, 60, 63, 65, 66, 67, 67, 68

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(18) = #{10+8x1, 18x1} = 2;
a(19) = #{19, 10+9x1, 19x1} = 3;
a(20) = #{19+1, 10+10, 10+10x1, 19x1} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.
A147706. [From Reinhard Zumkeller, Nov 11 2008]
A017173, A156144, A156145. [From Reinhard Zumkeller, Feb 05 2009]

a116371 n = p a017173_list n where
   p _  0 = 1
   p [] _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 04 2014

A116373 Number of partitions of n into parts with digital root = 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 5, 0, 0, 6, 0, 0, 7, 0, 0, 8, 0, 0, 10, 0, 0, 11, 0, 0, 13, 0, 0, 15, 0, 0, 17, 0, 0, 19, 0, 0, 23, 0, 0, 26, 0, 0, 29, 0, 0, 33, 0, 0, 38, 0, 0, 42, 0, 0, 48, 0, 0, 54, 0, 0, 61, 0, 0, 68, 0, 0, 77, 0, 0, 85, 0, 0, 96

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(21) = #{21, 12+3+3+3, 3+3+3+3+3+3+3} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

a(n) = A035382(floor(n/3))*0^(n mod 3).

A116376 Number of partitions of n into parts with digital root = 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 3, 0, 0, 2, 0, 0, 3, 0, 0, 3, 0, 0, 4, 0, 0, 4, 0, 0, 6, 0, 0, 5, 0, 0, 7, 0, 0, 7, 0, 0, 9, 0, 0, 9, 0, 0, 12, 0, 0, 11, 0, 0, 15, 0, 0, 15, 0, 0, 18, 0, 0, 19, 0, 0, 23, 0, 0, 23, 0, 0, 29, 0, 0, 29, 0, 0, 35, 0, 0, 37

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116377(n) - A116378(n) - A114099(n).

			a(30) = #{24+6, 15+15, 6+6+6+6+6} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

Cf. A147706.

N:= 1000: # to get a(1) to a(N)
g:= mul(1/(1-x^(6+9*j)), j=0..floor((N-6)/9)):
S:= series(g, x, N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 13 2015

a(n) = A035386(floor(n/3))*0^(n mod 3).

G.f.: Product_{j>=0} 1/(1 - x^(6+9*j)). - Robert Israel, Apr 13 2015

A116374 Number of partitions of n into parts with digital root = 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 3, 4, 1, 2, 3, 5, 2, 2, 3, 6, 4, 2, 3, 6, 6, 3, 3, 6, 7, 6, 3, 6, 8, 9, 4, 6, 8, 11, 7, 6, 8, 12, 11, 7, 8, 13, 14, 11, 8, 13, 16, 16, 9, 13, 17, 21, 13, 13, 18, 24, 20, 14, 18, 26

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116375(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(39) = #{31+4+4, 22+13+4, 13+13+13} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116375 Number of partitions of n into parts with digital root = 5.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 1, 2, 1, 1, 0, 2, 2, 1, 1, 1, 3, 2, 1, 1, 3, 3, 2, 1, 2, 4, 3, 2, 1, 4, 5, 3, 2, 2, 6, 5, 3, 2, 5, 7, 5, 3, 3, 8, 8, 5, 3, 6, 10, 8, 5, 4, 10, 11, 8, 5, 8, 13, 12, 8, 6, 13, 15, 12, 8, 10, 18, 16, 12, 9, 17, 21, 17

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116376(n) - A116377(n) - A116378(n) - A114099(n).

			a(42) = #{32+5+5, 23+14+5, 14+14+14} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116377 Number of partitions of n into parts with digital root = 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 0, 2, 0, 2, 1, 1, 1, 0, 2, 0, 3, 1, 3, 1, 1, 2, 0, 3, 1, 4, 1, 3, 2, 1, 3, 1, 5, 1, 5, 2, 4, 3, 2, 5, 1, 6, 2, 7, 3, 5, 5, 2, 7, 2, 9, 3, 9, 5, 6, 7, 3, 10, 3, 12, 5, 11, 7, 7, 11, 4, 14, 5, 16, 7, 14

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116378(n) - A114099(n).

			a(48) = #{34+7+7, 25+16+7, 16+16+16} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A116378 Number of partitions of n into parts with digital root = 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 0, 1, 1, 2, 3, 3, 1, 0, 0, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 5, 4, 1, 1, 1, 2, 3, 5, 6, 7, 4, 2, 1, 2, 3, 5, 7, 9, 8, 6, 2, 2, 3, 5, 7, 10, 11, 11, 6, 3, 3, 5, 7, 11, 13

Offset: 1

Reinhard Zumkeller, Feb 12 2006

a(n) = A114102(n) - A116371(n) - A116372(n) - A116373(n) - A116374(n) - A116375(n) - A116376(n) - A116377(n) - A114099(n).

			a(51) = #{35+8+8, 26+17+8, 17+17+17} = 3.

Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A147706. [From Reinhard Zumkeller, Nov 11 2008]

A147706 Number of partitions of n into parts having distinct digital roots (A010888).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 11, 15, 16, 20, 25, 28, 32, 39, 46, 50, 62, 66, 78, 93, 101, 112, 132, 150, 161, 192, 202, 232, 268, 287, 312, 361, 400, 425, 497, 516, 582, 658, 698, 748, 858, 932, 982, 1135, 1164, 1296, 1443, 1519, 1610, 1845, 1968, 2059, 2360, 2395

Offset: 1

Reinhard Zumkeller, Nov 11 2008

a(n) <= A000009(n).

Likely a duplicate of A114098. [From R. J. Mathar, Dec 13 2008]

			A000009(16) = 32, in which the following 4 partitions
contain parts with common digital roots:
12 + 3 + 1, 11 + 3 + 2, 10 + 5 + 1 and 10 + 3 + 2 + 1,
therefore a(16) = 32 - 4 = 28.

A114102, A116371, A116372, A116373, A116374, A116375, A116376, A116377, A116378, A114099.

cp's OEIS Frontend

A114099 Number of partitions of n into parts with digital root = 9.

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A114102 Number of partitions of n such that all parts of a partition have the same digital root.

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A116371 Number of partitions of n into parts with digital root = 1.

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A116373 Number of partitions of n into parts with digital root = 3.

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A116376 Number of partitions of n into parts with digital root = 6.

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A116374 Number of partitions of n into parts with digital root = 4.

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A116375 Number of partitions of n into parts with digital root = 5.

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A116377 Number of partitions of n into parts with digital root = 7.

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A116378 Number of partitions of n into parts with digital root = 8.

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