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Search: id:A130841
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| A130841 |
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Number of ways to write n as a sum of oterms, where an oterm is an ordered product of (1+oterm), sorted by size and an empty product has value 1. |
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+0
1
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| 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 12, 15, 17, 23, 23, 31, 31, 41, 44, 52, 52, 69, 73, 85, 91, 109, 109, 136, 136, 162, 170, 193, 199, 248, 248, 279, 291, 344, 344, 406, 406, 466, 493, 545, 545, 646, 655, 740, 763, 860, 860, 986, 1002, 1132, 1163, 1272, 1272, 1484
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Every oterm is at least 1 (implicit) and every 1+oterm is at least 2. Therefore to write 1 as a product of (1+oterms) can only be done as an empty product, which has value 1. Therefore a(1) = 1.
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FORMULA
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a(n) = sum over sequences (n_1,n_2,...,n_k) such that 2 <= n_1 <= n_2 <= ... <= n_k and n1*n2*...*nk=n of the product of j from 1 to k of a(n_j-1). The program, in J, implements this formula. (It works by factorizing n and then grouping the factors in all distinct ways. This J code handles the a(1) case without requiring any exception case.)
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EXAMPLE
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a(8)=5 because we can write 8 as one of (1+1+1+1+1+1+1+1), (1+1+1+1+(1+1)*(1+1)), (1+(1+1)*(1+1+1)), (1+1)*(1+1+1+1), (1+1)*(1+1)*(1+1)*(1+1)
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PROGRAM
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(J) belly =: ~. @ (i."1~) @ (#~ #: (i.@ ^~))
bell =: (<"1@belly@#) </.&.> <
bells =: [: ~. [: /:~&.> [: /:~&.>&.> bell
fax =: [: >&.> [: */&.>&.> [: bells q:
weird =: [: +/ [: > [: */&.> [: $:"0&.> [: <:&.> fax
w =: weird"0
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CROSSREFS
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Sequence in context: A097450 A062303 A050318 this_sequence A002095 A029017 A035371
Adjacent sequences: A130838 A130839 A130840 this_sequence A130842 A130843 A130844
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KEYWORD
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nonn
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AUTHOR
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Dan Brown (drdrlb(AT)rogers.com), Jul 19 2007, revised Nov 23 2007
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