A130841 - OEIS

%I #22 Jan 21 2021 22:10:55

%S 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,

%T 91,109,109,136,136,162,170,193,199,248,248,279,291,344,344,406,406,

%U 466,493,545,545,646,655,740,763,860,860,986,1002,1132,1163,1272,1272,1484

%N 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.

%C 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.

%C a(n) is also the number of non-isomorphic Gödel algebras of cardinality n. - _Diego Valota_, Jul 03 2019

%D 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.

%H Pietro Codara, Gabriele Maurina, and Diego Valota, <a href="https://doi.org/10.15439/2020F169">Computing Duals of Finite Gödel Algebras</a>, Proceedings of the Federated Conference on Computer Science and Information Systems, Annals of Computer Science and Information Science (2020) Vol. 21, 31-34.

%F 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.)

%e 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]

%o (J) belly =: ~. @ (i."1~) @ (#~ #: (i.@ ^~))

%o bell =: (<"1@belly@#) </.&.> <

%o bells =: [: ~. [: /:~&.> [: /:~&.>&.> bell

%o fax =: [: >&.> [: */&.>&.> [: bells q:

%o weird =: [: +/ [: > [: */&.> [: $:"0&.> [: <:&.> fax

%o w =: weird"0

%K nonn

%O 1,4

%A _Daniel R. L. Brown_, Jul 19 2007, revised Nov 23 2007