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A062051
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Number of partitions of n into parts which are powers of 3.
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13
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1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, 72, 81, 81, 81, 93, 93, 93, 105, 105, 105, 117, 117, 117, 132, 132, 132, 147, 147, 147, 162
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of different partial sums of 1+[1,*3]+[1,*3]+..., where [1,*3] means we can either add 1 or multiply by 3. E.g. a(6)=3 because we have 6=1+1+1+1+1+1=(1+1)*3=1*3+1+1+1 - Jon Perry (perry(AT)globalnet.co.uk), Jan 01 2004
Also number of partitions of n into distinct 3-smooth parts. E.g. a(10) = #{9+1, 8+2, 6+4, 6+3+1, 4+3+2+1} = #{9+1, 3+3+3+1, 3+3+1+1+1+1, 3+1+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1} = 5. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 07 2005
Starts to differ from A008650 at a(81). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2010]
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LINKS
| M. Latapy, Partitions of an integer into powers, DMTCS Proceedings AA (DM-CCG), 2001, 215-228.
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FORMULA
| a(n)=A005704([n/3]).
G.f.: product_{k>=0} 1/(1-x^(3^k)). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 31 2010]
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EXAMPLE
| a(4) = 2 and the partitions are 3+1, 1+1+1+1; a(9) = 5 and the partitions are 9; 3+3+3; 3+3+1+1+1; 3+1+1+1+1+1+1; 1+1+1+1+1+1+1+1+1.
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PROG
| (PARI) { n=15; v=vector(n); for (i=1, n, v[i]=vector(2^(i-1))); v[1][1]=1; for (i=2, n, k=length(v[i-1]); for (j=1, k, v[i][j]=v[i-1][j]+1; v[i][j+k]=v[i-1][j]*3)); c=vector(n); for (i=1, n, for (j=1, 2^(i-1), if (v[i][j]<=n, c[v[i][j]]++))); c } (Jon Perry)
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CROSSREFS
| A005704 with terms repeated 3 times. Cf. A000123, A018819.
Cf. A000009, A003586, A105420, A039966.
Cf. A018819, A023893, A105420, A106244, A131995.
Cf. A179051. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 27 2010]
Sequence in context: A076973 A008649 A008650 * A179269 A108711 A029059
Adjacent sequences: A062048 A062049 A062050 * A062052 A062053 A062054
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KEYWORD
| nonn
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AUTHOR
| Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 06 2001
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Jun 11 2001
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