A050318 a(n) is the number of ways to write n as an mterm, where an mterm is an unordered sum which is either 2, or 1 + an unordered product of mterms.

1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 12, 15, 17, 23, 23, 31, 31, 41, 44, 52, 52, 69, 72, 84, 90, 108, 108, 135, 135, 161, 169, 192, 198, 246, 246, 277, 289, 342, 342, 404, 404, 464, 491, 543, 543, 644, 650, 734, 757, 853, 853, 978, 994, 1123, 1154, 1262, 1262

Offset

2,4

Comments

P. Freyd (see link) writes: A050318 is "the number of isomorphism types of [equationally linear Heyting semi-lattices] of order n ..., albeit shifted by 1." And: "A050318 shifted by 1 is the number of isomorphism types of distributive lattices in which any element below a coprime is itself a coprime. Which ... is the same as the number of distributive lattices in which any element above a prime is itself a prime." - Peter Luschny, Nov 13 2018

Links

Andrew Howroyd, Table of n, a(n) for n = 2..10000

Peter Freyd, On the size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras, July 17 2017.

Formula

Shifts left under transform T where Ta has Dirichlet g.f. Product_{n>=1}(1/(1-1/n^s)^a(n)).

Example

The different ways of writing the numbers 2 through 7 as mterms are:
  2 = 2,
  3 = 1 + 2,
  4 = 1 + (1+2),
  5 = 1 + (1+1+2) = 1 + 2*2,
  6 = 1 + (1+1+1+2) = 1 + (1+2*2),
  7 = 1 + (1+1+1+1+2) = 1 + (1+1+2*2) = 1 + 2*(1+2).

Prog

(PARI) seq(n)={my(v=vector(n, i, i==1)); for(k=2, n, v=dirmul(v, vector(#v, i, my(e=valuation(i, k)); if(i==k^e, binomial(v[k-1] + e - 1, e), 0)))); v} \\ Andrew Howroyd, Nov 17 2018

Crossrefs

Cf. A001055, A050319, A050365, A050366, A067765.

Sequence in context: A180682 A227426 A229950 * A130841 A002095 A029017

Adjacent sequences: A050315 A050316 A050317 * A050319 A050320 A050321

Keyword

nonn,eigen,nice,easy

Author

Christian G. Bower, Sep 15 1999

Status

approved