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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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. Prod_{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

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Last modified July 17 17:14 EDT 2019. Contains 325107 sequences. (Running on oeis4.)