A130841 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.

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

Offset: 1

Author

Daniel R. L. Brown, Jul 19 2007, revised Nov 23 2007

Keywords

• nonn

Comments

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.

a(n) is also the number of non-isomorphic Gödel algebras of cardinality n. - Diego Valota, Jul 03 2019

Examples

			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). [corrected by _Diego Valota_, Jul 03 2019]
		

References

• Diego Valota (2019) Spectra of Gödel Algebras. In: Silva A., Staton S., Sutton P., Umbach C. (eds) Language, Logic, and Computation. TbiLLC 2018. Lecture Notes in Computer Science, vol 11456. Springer, Berlin, Heidelberg.

Links

• Pietro Codara, Gabriele Maurina, and Diego Valota, Computing Duals of Finite Gödel Algebras, Proceedings of the Federated Conference on Computer Science and Information Systems, Annals of Computer Science and Information Science (2020) Vol. 21, 31-34.

Programs

J

belly =: ~. @ (i."1~) @ (#~ #: (i.@ ^~))
bell =: (<"1@belly@#)  <
bells =: [: ~. [: /:~&.> [: /:~&.>&.> bell
fax =: [: >&.> [: */&.>&.> [: bells q:
weird =: [: +/ [: > [: */&.> [: $:"0&.> [: <:&.> fax
w =: weird"0

Formula

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.)