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%I #19 Nov 17 2018 16:23:43
%S 1,1,1,1,1,2,2,3,3,4,4,6,6,8,9,11,11,15,15,19,21,25,25,33,33,39,42,50,
%T 50,63,63,74,78,89,91,110,110,125,131,152,152,181,181,206,217,242,242,
%U 285,286,322,333,372,372,428,432,486,501,551,551,636,636,699,724,799
%N a(n) is the number of ways to write n as an identity mterm, where an identity mterm is an unordered sum which is either 2, or 1 + an unordered product of distinct identity mterms.
%C P. Freyd (see link) writes: "Bower’s sequence A050365 can be interpreted to count the number of anti-symmetric elhsls (and elhas), ... those for which all automorphisms are one." Here 'elha' stands for 'equationally linear Heyting algebra' and 'elhsl' means 'equationally linear Heyting semi-lattice'. Then Freyd gives another interpretation and coins the name Bower-rank. "For a set of finite rank we may define its Bower-rank as the first ordinal larger than the product of the Bower-ranks of its elements. Then A050365 counts the number of sets of given Bower-rank." - _Peter Luschny_, Nov 13 2018
%H Andrew Howroyd, <a href="/A050365/b050365.txt">Table of n, a(n) for n = 2..10000</a>
%H Peter Freyd, <a href="https://www.math.upenn.edu/~pjf/Heyting.pdf">On the size of Heyting Semi-Lattices and Equationally Linear Heyting Algebras</a>, July 17 2017.
%F Shifts left under transform T where Ta has Dirichlet g.f. Product_{n>=1}(1+1/n^s)^a(n).
%e The different ways of writing the numbers 2 through 7 as identity mterms are:
%e 2 = 2,
%e 3 = 1 + 2,
%e 4 = 1 + (1+2),
%e 5 = 1 + (1+1+2),
%e 6 = 1 + (1+1+1+2),
%e 7 = 1 + (1+1+1+1+2) = 1 + 2*(1+2).
%o (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), 0)))); v} \\ _Andrew Howroyd_, Nov 17 2018
%Y Cf. A045778, A050318, A050319, A050366, A067765.
%K nonn,eigen
%O 2,6
%A _Christian G. Bower_, Oct 15 1999
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