A073524 -id:A073524 - cp's OEIS frontend

A081851 Consider recurrence b(0) = (2n+1)/4, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached (or -1 if no integer is ever reached).

5, 7, 36, 1711985, 13, 15, 1700, 114, 168, 42000323, 275, 324, 58, 62, 23658393, 6055, 58311963, 9321, 121770, 13760, 135, 141, 1960, 344148, 5734229, 3391007266515, 8825709, 23546737390632357, 244, 252, 1526332099115586230, 105432399233, 27538521, 5680

Offset: 2

N. J. A. Sloane, Apr 13 2003

nonn

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A073524, A081849, A081850.

g:= proc(n) local b0, b, count;
  b0:= (2*n+1)/4; b:= b0;
  for count from 1 do
    b:= b0 * ceil(b);
    if b::integer then return b fi
  od
end proc:
map(g, [$2..100]); # Robert Israel, Sep 21 2018

A075107 Number of steps to reach the first integer (= A075108(n)) starting with n/floor(log_2(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 0, 2, 0, 1, 1, 0, 5, 2, 0, 3, 2, 0, 0, 3, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 2, 5, 3, 0, 16, 2, 4, 6, 0, 4, 16, 9, 2, 0, 1, 1, 1, 1, 0, 4, 2, 4, 8, 0, 2, 5, 2, 8, 0, 9, 3, 4, 5, 2, 0, 1, 1, 1, 1, 1, 0, 5, 7, 3, 2, 4, 0, 9, 4, 1, 2, 3, 0, 4, 1, 2, 1

Offset: 2

Reinhard Zumkeller, Sep 02 2002

nonn

Starting values given by A075105(n)/A075106(n).

			a(5)=2 since 5/floor(log_2(5)) = 5/2 -> 15/2 -> 60 = A075108(5).

Cf. A068119, A073524, A075102, A075120.

More terms from Jinyuan Wang, Jan 15 2022

A075120 Number of steps to reach the first integer (= A075121(n)) starting with n/floor(sqrt(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 1, 0, 0, 5, 2, 0, 3, 2, 0, 0, 3, 3, 2, 0, 2, 1, 3, 0, 0, 8, 3, 5, 2, 0, 3, 2, 5, 3, 0, 0, 6, 2, 2, 4, 2, 0, 5, 3, 1, 4, 2, 0, 0, 7, 12, 5, 11, 12, 2, 0, 4, 3, 9, 12, 6, 16, 0, 0, 3, 7, 8, 4, 4, 4, 2, 0, 2, 4, 5, 1, 5, 2, 5, 0, 0, 7, 3, 22, 3, 3

Offset: 1

Reinhard Zumkeller, Sep 03 2002

nonn

Starting with A073890(n)/A075119(n).

			a(13)=3 since 13/floor(sqrt(13)) = 13/3 -> 65/3 -> 1430/3 -> 227370 = A075121(13).

Cf. A068119, A073524, A075102, A075107, A075121.

More terms from Jinyuan Wang, Jan 15 2022

A074735 Number of steps to reach an integer starting with (n+3)/4 and iterating the map x -> x*ceiling(x).

Original entry on oeis.org

0, 3, 1, 2, 0, 3, 2, 8, 0, 1, 1, 1, 0, 3, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 0, 7, 4, 4, 0, 4, 1, 2, 0, 4, 2, 3, 0, 1, 1, 1, 0, 2, 3, 4, 0, 2, 1, 8, 0, 4, 2, 3, 0, 1, 1, 1, 0, 6, 5, 4, 0, 3, 1, 2, 0, 5, 2, 4, 0, 1, 1, 1, 0, 5, 3, 2, 0, 2, 1, 3, 0, 2, 2, 2, 0, 1, 1, 1, 0, 4, 4, 5, 0, 6, 1, 2, 0, 3, 2, 5, 0

Offset: 1

Benoit Cloitre, Sep 05 2002

nonn

Let S(n) = Sum_{k=1..n} a(k) then it seems that S(n) is asymptotic to 2n. S(n)=2n for many values of n, namely n=10,128,198,199,237,238,241,242,246,247,249,267,329... More generally, starting with (n+2^m-1)/2^m and iterating the same map seems to produce the same kind of behavior for a(n) (i.e., Sum_{k=1..n} a(k) is asymptotic to c(m)*n where c(m) depends on m and c(m) is a power of 2).

Cf. A073524, A073341, A068119.

Table[Length[NestWhileList[# Ceiling[#]&,(n+3)/4,!IntegerQ[#]&]]-1,{n,110}] (* Harvey P. Dale, Apr 11 2020 *)

a(n)=if(n<0,0,s=(n+3)/4; c=0; while(frac(s)>0,s=s*ceil(s); c++); c)

Special cases: for k>= 0 a(4k+1) = 0, a(16k+10) = a(16k+11) = a(16k+12) = 1.

Offset corrected by Sean A. Irvine, Jan 25 2025

A074923 Number of steps to reach the first integer starting with prime(n+1)/prime(n) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

1, 6, 18, 5, 5, 7, 37, 132, 29, 63, 46, 11, 7, 212, 25, 38, 145, 136, 46, 118, 39, 37, 24, 15, 19, 85, 211, 179, 249, 266, 190, 87, 259, 216, 16, 70, 210, 56, 255, 94, 143, 244, 482, 558, 46, 327, 271, 93, 110, 8, 121, 29, 421, 329, 47, 40, 394, 376, 64, 249, 49, 390, 321

Offset: 1

Reinhard Zumkeller, Oct 04 2002

nonn

			a(4)=5 since A000040(4+1)/A000040(4)=11/7 -> 22/7 -> 88/7 -> 1144/7 -> 187616/7 -> 718381664.

Sean A. Irvine, Table of n, a(n) for n = 1..200

Cf. A073524, A000040.

Extended by Max Alekseyev, Jul 28 2009

A075428 Number of steps to reach the first integer (= A075429(n)) starting with 1 + 2/(n(n+1)) and iterating the map x -> xceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 2, 2, 7, 4, 3, 4, 8, 7, 41, 15, 15, 18, 10, 8, 17, 10, 12, 10, 5, 9, 9, 10, 11, 14, 39, 12, 33, 65, 9, 31, 6, 23, 26, 6, 71, 98, 22, 43, 20, 26, 4, 33, 12, 36, 128, 43, 18, 10, 81, 28, 54, 24, 12, 8, 38, 19, 10, 41, 36, 83, 32, 9, 9, 12, 13, 60, 58, 19, 32

Offset: 1

Reinhard Zumkeller, Sep 16 2002

nonn

			a(5)=4 since 1 + 2/(5*6) = 1 + 1/15 = 16/15 -> 32/15 -> 32/5 -> 224/5 -> 2016=A075429(5).

Cf. A000217, A073524, A075429.

More terms from Jinyuan Wang, Jan 15 2022

A081852 Consider recurrence b(0) = n/3, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached (or -1 if no integer is ever reached).

Original entry on oeis.org

1, 4, 20, 2, 7, 8, 3, 26631380, 55, 4, 416, 112, 5, 32, 34, 6, 285, 13960, 7, 67358874, 214544, 8, 75, 78, 9, 62186796, 7399041846, 10, 1178, 173857344, 11, 136, 140, 12, 24494, 2090, 13, 78824360, 2624, 14, 215, 220, 15, 3772, 61617, 16, 23001295794169, 78900, 17, 312

Offset: 3

N. J. A. Sloane, Apr 13 2003

nonn

Cf. A073524, A081849, A081850, A081851. A074078 gives number of steps to reach an integer.

A118020 Number of steps to reach an integer starting with (n+1)/n and using the approximate cubing map x -> x*ceiling(x^2); or -1 if no integer is ever reached.

Original entry on oeis.org

0, -1, 3, -1, 12, 2, 3, -1, 7, 2, 18, 8, 7, 22, 3, -1, 6, 8, 17, -1, 25, 3, 68, 4, 8, 14, 3, 11, 171, 6, 19, 5, 11, 11, 16, 6, 23, 19, 6, -1, 55, 3, 23, 10, 4, 26, 58, 6, 12, 3, 13, 3, 62, 9, 9, 4, 19, 62, 105, 9, 7, 24, 7, -1, 3, 17, 16, 12, 66, 21, 66, -1, 63, 65, 6, 28, 20, 20, 54, -1, 13, 92, 19, 21, 7, 9, 34, 36, 67, 5, 20, 5, 29, 62, 39, 6, 105

Offset: 1

T. D. Noe, Apr 10 2006

sign

This sequence is similar to A073524, approximate squaring. However, for the cubing map it is easy to show that fractions of the form odd/2 never yield an integer. Hence if an iterate ever has this form, then we know it will never yield an integer. The computations, similar to A073524, must be done modulo n^max for some max > 2*a(n)+2.

			a(3)=3 because 4/3 -> 8/3 -> 64/3 -> 9728.

Cf. A118021 (n for which a(n)=-1).

Stuck[x_] := OddQ[Numerator[x]] && (Denominator[x]==2); Table[lim=50; While[k=0; x=1+1/n; m=n^lim; While[2k=lim-3, lim=2*lim]; If[Stuck[x],-1,k], {n,200}]

A074212 Number of steps to reach an integer starting with (2^n + 1)/2^n and iterating the map x->x*ceiling(x).

Original entry on oeis.org

1, 3, 4, 8, 6, 6, 14, 14, 15, 9, 12, 20, 21, 21, 22, 30, 28, 33, 22, 11, 16, 34, 31, 23, 32, 30, 25, 43, 32, 29, 35, 40, 31, 58, 49, 47, 39, 43, 45, 46, 39, 44, 32, 44, 22, 56, 51, 61, 48, 46, 55, 68, 69, 60, 69, 70, 78, 89, 72, 93, 61, 64, 80, 71, 60, 58, 71

Offset: 1

Benoit Cloitre, Sep 17 2002

nonn

Is a(n) > n for n > 10? Does lim_{n->infinity} a(n)/n exist?

a(n) <= n for n = 1, 6, 10, 20, 21, 24, 27, ... - Amiram Eldar, Nov 28 2020

Cf. A073524.

a[n_] := Module[{x = (2^n + 1)/2^n, nstep = 0}, While[!IntegerQ[x], nstep++; x *= Ceiling[x]]; nstep]; Array[a, 15] (* Amiram Eldar, Nov 28 2020 *)

a(16)-a(17) from Ryan Propper, Mar 18 2008
a(18)-a(21) from Lars Blomberg, Jan 17 2013
a(22)-a(27) from Amiram Eldar, Nov 28 2020
More terms from Jinyuan Wang, Jan 15 2022

A074970 Number of steps to reach the first integer starting with sigma(n)/n and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

Original entry on oeis.org

0, 1, 2, 2, 18, 0, 3, 2, 3, 3, 26, 1, 9, 3, 7, 3, 6, 3, 56, 10, 9, 15, 42, 2, 10, 3, 3, 0, 38, 17, 79, 3, 27, 16, 8, 8, 200, 76, 4, 3, 36, 2, 4, 15, 4, 51, 11, 2, 4, 6, 22, 42, 43, 5, 25, 2, 4, 103, 37, 17, 18, 20, 8, 3, 22, 25, 225, 59, 6, 13, 20, 4, 182, 13, 8

Offset: 1

Reinhard Zumkeller, Oct 05 2002

nonn

a(p) = A073524(p) for primes p.

			a(26)=3 since (26+13+2+1)/26=21/13 -> 42/13 -> 168/13 -> 168.

Cf. A000203, A009194, A073524.

a(20) corrected by and more terms from Jinyuan Wang, Jan 15 2022

cp's OEIS Frontend

A081851 Consider recurrence b(0) = (2n+1)/4, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached (or -1 if no integer is ever reached).

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Maple

A075107 Number of steps to reach the first integer (= A075108(n)) starting with n/floor(log_2(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A075120 Number of steps to reach the first integer (= A075121(n)) starting with n/floor(sqrt(n)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A074735 Number of steps to reach an integer starting with (n+3)/4 and iterating the map x -> x*ceiling(x).

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Mathematica

PARI

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A074923 Number of steps to reach the first integer starting with prime(n+1)/prime(n) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A075428 Number of steps to reach the first integer (= A075429(n)) starting with 1 + 2/(n*(n+1)) and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A081852 Consider recurrence b(0) = n/3, b(n) = b(0)*ceiling(b(n-1)); sequence gives first integer reached (or -1 if no integer is ever reached).

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A118020 Number of steps to reach an integer starting with (n+1)/n and using the approximate cubing map x -> x*ceiling(x^2); or -1 if no integer is ever reached.

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Mathematica

A074212 Number of steps to reach an integer starting with (2^n + 1)/2^n and iterating the map x->x*ceiling(x).

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Mathematica

Extensions

A074970 Number of steps to reach the first integer starting with sigma(n)/n and iterating the map x -> x*ceiling(x), or -1 if no integer is ever reached.

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A075428 Number of steps to reach the first integer (= A075429(n)) starting with 1 + 2/(n(n+1)) and iterating the map x -> xceiling(x), or -1 if no integer is ever reached.