A354459 Lazy cutter's sequence (see Comments).

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

Offset

1,1

Comments

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).

Links

Table of n, a(n) for n=1..86.

Formula

h(n) = A182972(n)/A182973(n) = b/c, c = x*b + r and a(n) = (x+1)*b + r - 1.

Example

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.

Mathematica

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

Crossrefs

Cf. A023022, A092542, A092543, A182972, A182973.

Sequence in context: A335572 A268680 A126974 * A089058 A282717 A187103

Adjacent sequences: A354456 A354457 A354458 * A354460 A354461 A354462

Keyword

nonn

Author

Ivan N. Ianakiev, May 31 2022

Status

approved