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A354459
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Lazy cutter's sequence (see Comments).
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0
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2, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23
(list;
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refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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From the infinite sequence G of fractions that may be used to demonstrate the countability of rational numbers, where a(n) = A092542(n)/A092543(n), form a new sequence H by taking only those terms of G that are proper fractions unequal to a fraction that appears earlier in H (making H the list of all proper fractions without repetitions). Let b/c be the n-th term of H and b be the number of congruent pizzas that have to be equally divided between c people by means of radial cuts. a(n) is the minimum number of cuts to achieve such a division.
H can be directly calculated as its n-th term equals A182972(n)/A182973(n). H starts with 1/2, 1/3, 1/4, 2/3, 1/5, 1/6, 2/5, 3/4, 3/5, 1/7, 1/8, 2/7, 4/5, 3/7, 1/9.
As a(n) is equal for all proper fractions b/c such that b + c = n, counting the number of equal consecutive terms of this sequence gives A023022 from its third term onwards (see Geoffrey Critzer's and Reinhard Zumkeller's comments at A023022).
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LINKS
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FORMULA
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h(n) = A182972(n)/A182973(n) = b/c, c = x*b + r and a(n) = (x+1)*b + r - 1.
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EXAMPLE
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To equally divide 4 pizzas between 7 people we can divide each pizza into 7 equal parts with 7 radial cuts making the total number of cuts 28 (far from minimal). Ancient Egyptians, representing 4/7 as 1/2 + 1/14, would cut all pizzas into halves (8 cuts) and one of the halves into 7 equal pieces (6 additional cuts), making the total number of cuts 8 + 6 = 14. We can do even better by cutting each pizza into two pieces (3/7 and 4/7), for a total of 8 cuts, and dividing one 3/7 piece to 3 equal pieces (2 additional cuts), minimizing the total number of cuts to 8 + 2 = 10. Since the 19th term of H sequence is 4/7, a(19) = 10.
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MATHEMATICA
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a092542=Flatten[Table[Join[Range[2n-1], Reverse@Range[2n-2]], {n, 12}]];
a092543=Take[Cases[Import["https://oeis.org/A092543/b092543.txt", "Table"], {_, _}][[All, 2]], 276]; g=a092542/a092543; h=DeleteDuplicates[Select[g, #<1&]];
a[n_]:=Module[{x=Floor[Denominator[n]/Numerator[n]], r=Mod[Denominator[n],
Numerator[n]]}, (x+1)*Numerator[n]+r-1]; a/@h
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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