A097051 -id:A097051 - cp's OEIS frontend

A097053 First occurrence of n in A097051.

1, 2, 3, 8, 10, 12, 14, 32, 36, 40, 44, 48, 52, 56, 60, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 512, 528, 544, 560, 576, 592, 608, 624, 640, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 816, 832, 848, 864, 880, 896

Offset: 1

Reinhard Zumkeller and Robert G. Wilson v, Jul 21 2004

nonn

Cf. A097051.

a[1] = 1; a[n_] := a[n] = Floor[ n / a[ Floor[n / 2]]]; t = Table[ a[n], {n, 1000}]; b[n_] := Block[{p = Position[t, n, 1, 1]}, If[p == {}, 0, p]]; Flatten[ Table[ b[n], {n, 60}]]

For n>1, a(n) = A053644(n) * n / 2.

Formula added by Max Alekseyev, Mar 02 2011

A096036 a(n) = ceiling(n/a(ceiling(n/2))); a(1) = 1.

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 6, 6, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 6, 6, 6, 6, 8, 8, 8, 8, 7, 7, 8, 8, 8, 8, 8, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 11, 11, 12, 12, 12, 12, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 11, 11, 11, 11, 12, 12, 12

Offset: 1

Reinhard Zumkeller, Jul 20 2004

nonn

			a(50)=ceiling(50/a(25))
..... a(25)=ceiling(25/a(13))
........... a(13)=ceiling(13/a(7))
................. a(7)=ceiling(7/a(4))
...................... a(4)=ceiling(4/a(2))
........................... a(2)=ceiling(2/a(1))
................................ a(1)=1
........................... a(2)=ceiling(2/a(1))=ceiling(2/1)=2
...................... a(4)=ceiling(4/a(2))=ceiling(4/2)=2
................. a(7)=ceiling(7/a(4))=ceiling(7/2)=4
........... a(13)=ceiling(13/a(7))=ceiling(13/4)=4
..... a(25)=ceiling(25/a(13))=ceiling(25/4)=7
a(50)=ceiling(50/a(25))=ceiling(50/7)=8.

Cf. A097051, A097052.

a[1] = 1; a[n_] := a[n] = Ceiling[ n / a[Ceiling[ n/2]]]; Table[ a[n], {n, 91}] (* Robert G. Wilson v, Jul 21 2004 *)

A372970 a(1)=1, then a(n) = floor(n/max(a(n-1),a(floor(n/2)))).

Original entry on oeis.org

1, 2, 1, 2, 2, 3, 2, 4, 2, 5, 2, 4, 3, 4, 3, 4, 4, 4, 4, 4, 4, 5, 4, 6, 4, 6, 4, 7, 4, 7, 4, 8, 4, 8, 4, 9, 4, 9, 4, 10, 4, 10, 4, 8, 5, 9, 5, 8, 6, 8, 6, 8, 6, 9, 6, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 8, 8, 9, 8, 8, 8, 10, 8, 8, 8, 10, 8, 11, 8, 11, 8, 10, 9, 10, 9, 10, 9, 10, 9, 11

Offset: 1

Benoit Cloitre, May 18 2024

nonn

It seems that limsup and liminf of a(n)/sqrt(n) exist (see link).

Benoit Cloitre, Plot of a(n)/sqrt(n) for n=1 up to 400000.
Hugo Pfoertner, Plot of a(n)/sqrt(n), n=1..400000, zoom into pdf to see details.

Cf. A372971, A097051.

a(n)=if(n<2,1,floor(n/max(a(n-1),a(n\2))))

A372971 a(1)=1, then a(n) = floor(n/min(a(n-1),a(floor(n/2)))).

Original entry on oeis.org

1, 2, 3, 2, 2, 3, 2, 4, 4, 5, 5, 4, 4, 7, 7, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 6, 6, 4, 7, 4, 7, 8, 8, 8, 8, 9, 9, 9, 9, 8, 8, 10, 10, 8, 9, 11, 11, 8, 8, 8, 8, 8, 8, 9, 9, 14, 14, 8, 8, 15, 15, 8, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 8, 9, 8, 9, 8, 9, 8, 10

Offset: 1

Benoit Cloitre, May 18 2024

It seems that limsup and liminf of a(n)/sqrt(n) exist (see link).

Hugo Pfoertner, Table of n, a(n) for n = 1..10000
Benoit Cloitre, Plot of a(n)/sqrt(n) for n=1 up to 400000
Hugo Pfoertner, Plot of a(n)/sqrt(n), n=1..400000, zoom into pdf to see details.

Cf. A372970, A097051.

a[1]=1; a[n_]:=Floor[n/Min[a[n-1],a[Floor[n/2]]]]; Array[a,80] (* Stefano Spezia, May 18 2024 *)

a(n)=if(n<2,1,floor(n/min(a(n-1),a(n\2))))

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A097053 First occurrence of n in A097051.

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A096036 a(n) = ceiling(n/a(ceiling(n/2))); a(1) = 1.

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A372970 a(1)=1, then a(n) = floor(n/max(a(n-1),a(floor(n/2)))).

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A372971 a(1)=1, then a(n) = floor(n/min(a(n-1),a(floor(n/2)))).

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