cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Daniel R. L. Brown

Daniel R. L. Brown's wiki page.

Daniel R. L. Brown has authored 2 sequences.

A162599 Sum of digits (bits) in base two representation of 3^(2^n).

Original entry on oeis.org

2, 2, 3, 6, 11, 26, 56, 97, 204, 415, 769, 1606, 3268, 6460, 12889, 25915, 52046, 104099, 207000, 416542, 831430, 1662247, 3324374, 6646337, 13292865, 26593542, 53182276, 106364450, 212732814, 425465107, 850887793, 1701801109
Offset: 0

Views

Author

Daniel R. L. Brown, Jul 07 2009

Keywords

Examples

			3^(2^2) = 81 = 64 + 16 + 1, so a(2) = 3.

Crossrefs

Cf. A000120, A011764 (3^2^n).

Programs

  • Maple
    a := proc (n) local b2: b2 := convert(3^(2^n), base, 2): add(b2[j], j = 1 .. nops(b2)) end proc: seq(a(n), n = 0 .. 20); # Emeric Deutsch, Jul 21 2009
  • Mathematica
    Table[Total[IntegerDigits[3^2^n,2]],{n,0,25}] (* The program generates the first 26 terms of the seequence. *) (* Harvey P. Dale, Jun 16 2024 *)
  • PARI
    a(n) = hammingweight(3^(2^n)); \\ Michel Marcus, Mar 03 2019

Extensions

Extended by Emeric Deutsch, Jul 21 2009
More terms from Michel Marcus, Mar 03 2019

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.

Original entry on oeis.org

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

Views

Author

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

Keywords

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.)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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