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A050365
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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.
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4
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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, 50, 63, 63, 74, 78, 89, 91, 110, 110, 125, 131, 152, 152, 181, 181, 206, 217, 242, 242, 285, 286, 322, 333, 372, 372, 428, 432, 486, 501, 551, 551, 636, 636, 699, 724, 799
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OFFSET
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2,6
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COMMENTS
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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
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LINKS
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FORMULA
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Shifts left under transform T where Ta has Dirichlet g.f. Product_{n>=1}(1+1/n^s)^a(n).
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EXAMPLE
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The different ways of writing the numbers 2 through 7 as identity mterms are:
2 = 2,
3 = 1 + 2,
4 = 1 + (1+2),
5 = 1 + (1+1+2),
6 = 1 + (1+1+1+2),
7 = 1 + (1+1+1+1+2) = 1 + 2*(1+2).
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,eigen
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AUTHOR
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STATUS
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approved
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