A090895 -id:A090895 - cp's OEIS frontend

A185038 Positions at which A090895 (a(1) = 1 then a(n) = a(n-1)/2 if a(n-1) is even, a(n) = a(n-1) + n otherwise) reach the value of 1.

1, 8, 15, 64, 7069, 229826, 62906548, 104509874538, 209233407362, 474536678820, 2653613575299, 14802269029898

Offset: 1

Sergio Pimentel, Mar 01 2012

It seems that this sequence is infinite.

			Example: the sequence a(1)=1 then a(n)=a(n-1)/2 if a(n-1) is even, a(n) = a(n-1) + n otherwise goes 1, 3, 6, 3, 8, 4, 2, 1, 10, 5, 16, 8, 4, 2, 1, etc... which has ones at 1, 8, 15, 64, etc...

Cf. A090895.

import Data.List (elemIndices)
a185038 n = a185038_list !! (n-1)
a185038_list = map (+ 1) $ elemIndices 1 a090895_list
-- Reinhard Zumkeller, Mar 02 2012

a(8)-a(12) from Hiroaki Yamanouchi, Oct 04 2014

A208852 Smallest m such that A090895(m) = n.

Original entry on oeis.org

1, 7, 2, 6, 10, 3, 48, 5, 21, 9, 44, 81, 27, 47, 55, 11, 16, 20, 1058, 364, 745, 43, 300, 80, 69, 26, 24, 46, 5901, 54, 22, 59, 32, 17, 30, 19, 568076, 1057, 75, 363, 28, 744, 87, 42, 169, 299, 40, 79, 315, 68, 186, 25, 101, 23, 38, 45, 2924, 5900, 413, 53

Offset: 1

Reinhard Zumkeller, Mar 02 2012

nonn

A090895(a(n)) = n and A090895(m) <> n for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1097

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a208852 = (+ 1) . fromJust . (`elemIndex` a090895_list)

A135287 a(0)=1; for n > 0, a(n) = a(n-1)+n if a(n-1) is odd, else a(n) = a(n-1)/2.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 7, 14, 7, 16, 8, 4, 2, 1, 15, 30, 15, 32, 16, 8, 4, 2, 1, 24, 12, 6, 3, 30, 15, 44, 22, 11, 43, 76, 38, 19, 55, 92, 46, 23, 63, 104, 52, 26, 13, 58, 29, 76, 38, 19, 69, 120, 60, 30, 15, 70, 35, 92, 46, 23, 83

Offset: 0

Ctibor O. Zizka, Dec 03 2007, Dec 05 2007

Let a(0), C1, C2, C be integers. Consider the sequence a(n) = a(n-1) + C1*n + C2 if a(n-1) is not divisible by C or a(n) = a(n-1)/C otherwise.
For a fixed C1, C2, C this sequence shows chaotic behavior for some a(0) and a highly regular behavior for other a(0).
The parameter C1 tells how many regular subclasses are there.
The sequence grows roughly as a(n) ~ n*const.
Here C = 2. Other sequences showing very interesting behavior have C = power of 2.
Example: C1=3, C2=10, C=3. Thus a(n)= a(n-1)+3*n+10 if a(n-1) is not divisible by 3, or a(n)= a(n-1)/3 otherwise. There are 2 classes:
a regular class with 3 subclasses (C1=3) for initial values
{a(0)=3,38,79,...}
{a(0)=1,8,12,42,47,49,63,77,88,...}
{a(0)=2,43,45,...}
and a "chaotic" class for other initial values a(0).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A008336, A005132, A135294.

Cf. A090895.

a135287 n = a135287_list !! n
a135287_list = 1 : f 1 1 where
   f x y = z : f (x + 1) z where
        z = if m == 0 then y' else x + y; (y',m) = divMod y 2
-- Reinhard Zumkeller, Mar 02 2012

A135287 := proc(n) option remember ; if n = 0 then 1 ; elif A135287(n-1) mod 2 = 0 then A135287(n-1)/2 ; else n+A135287(n-1) ; fi ; end: seq(A135287(n),n=0..60) ; # R. J. Mathar, Dec 12 2007

nxt[{n_,a_}]:={n+1,If[OddQ[a],a+n+1,a/2]}; NestList[nxt,{0,1},60][[;;,2]] (* Harvey P. Dale, Mar 02 2023 *)

More terms from R. J. Mathar, Dec 12 2007

Offset fixed by Reinhard Zumkeller, Mar 02 2012

A208884 a(n) = (a(n-1) + n)/2^k where 2^k is the largest power of 2 dividing a(n-1) + n, for n>1 with a(1)=1.

Original entry on oeis.org

1, 3, 3, 7, 3, 9, 1, 9, 9, 19, 15, 27, 5, 19, 17, 33, 25, 43, 31, 51, 9, 31, 27, 51, 19, 45, 9, 37, 33, 63, 47, 79, 7, 41, 19, 55, 23, 61, 25, 65, 53, 95, 69, 113, 79, 125, 43, 91, 35, 85, 17, 69, 61, 115, 85, 141, 99, 157, 27, 87, 37, 99, 81, 145, 105, 171

Offset: 1

Paul D. Hanna, Mar 02 2012

nonn

In other words, to get a(n), add n to a(n-1) and compute the odd part (A000265) of the sum. - Ralf Stephan, Oct 27 2013
POSITIONS of odd numbers in the initial 7000000 terms begin:
1: [1, 7, 69, 285, 3601, 5167, 92989, 112651, 6933175, ...];
3: [2, 3, 5, 613, 8461, 46749, 81237, 102171, 126661, 3309589, ...];
5: [13, 97, 2431, 92095, ...];
7: [4, 33, 3167, 78095, 2723179, ...];
9: [6, 8, 9, 21, 27, 303, 2017, 3239, 3765, 6753, 28387, 251451, ...];
11: [75, 15823, 28221, 4091959, 5820487, ...];
13: [22975, 42391, 3729249, ...];
15: [11, 22587, 2527579, 6954893, ...];
17: [15, 51, 3121, 13433, 74763, 376853, 576439, 896899, ...];
19: [10, 14, 25, 35, 291, 77747, 757319, 1227595, 2307099, ...];
21: [1417, 1557, 712229, 2563807, ...];
23: [37, 127, 609, 2211, 5563, 199901, ...];
25: [17, 39, 221, 1145, 3425, 17593, 4318897, ...];
27: [12, 23, 59, 73, 289, 1149, 3393, 20439, 37107, ...];
29: [573, 33315, 61505, 467047, 491359, 1170709, 1492309, 2498593, 3017011, ...];
31: [19, 22, 229, 409, 6199, 60529, 3602675, 4108215, 4604929, ...]; ...
From Ya-Ping Lu, Jun 25 2020: (Start)
Conjecture: For any given odd number m, there exists a number n_max such that all odd numbers <= m can be found in the sequence a(n) with n <= n_max. For example:
m = 1, n_max = 1;
m = 3, n_max = 2;
m = 5, n_max = 13;
m = 11, n_max = 75
m = 13, n_max = 22975;
m = 305, n_max = 1025715;
m = 749, n_max = 14695985;
m = 795, n_max = 150788015;
m = 7525, n_max = 31129547917;
...
If the conjecture above is true, this sequence contains all odd numbers. (End)

			a(2) = 1 + 2 = 3;
a(3) = (3 + 3)/2 = 3;
a(4) = 3 + 4 = 7;
a(5) = (7 + 5)/4 = 3;
a(6) = 3 + 6 = 9;
a(7) = (9 + 7)/16 = 1; ...

Paul D. Hanna, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of the parity of n)

Cf. A069834, A090895, A114216, A335817.

a[1]=1; a[n_] := a[n] = #/2^IntegerExponent[#, 2] &@ (n + a[n-1]); Array[a, 70] (* Giovanni Resta, Jun 25 2020 *)

{a(n)=if(n==1, 1, (a(n-1)+n)/2^valuation(a(n-1)+n,2))}

{A=vector(1024); a(n)=A[n]=if(n==1, 1, (A[n-1]+n)/2^valuation(A[n-1]+n,2))}
for(n=1,#A,print1(a(n),", "))

A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 7, 2, 1, 10, 5, 8, 4, 2, 1, 16, 8, 4, 2, 1, 21, 2, 1, 24, 12, 6, 3, 26, 13, 18, 9, 24, 12, 6, 3, 34, 17, 22, 11, 30, 15, 28, 14, 7, 39, 8, 4, 2, 1, 50, 25, 28, 14, 7, 49, 8, 4, 2, 1, 60, 30, 15, 49, 16, 8, 4, 2, 1, 69, 2, 1, 72, 36, 18, 9, 68, 34, 17, 63, 18, 9, 74, 37, 48, 24, 12, 6, 3

Offset: 1

Altug Alkan, Jun 28 2021

Let a_i(1) = 1 and a_i(n) = a_i(n-1)/(i+1) if a_i(n-1) is divisible by i+1, otherwise a_i(n) = n - a_i(n-1). This sequence is a_1(n) and A345886 is a_2(n).

Conjecture: a_i(n) hits every positive integers infinitely many times for all i >= 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A090895, A135287, A345886.

a[1] = 1; a[n_] := a[n] = If[EvenQ[a[n - 1]], a[n - 1]/2, n - a[n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 29 2021 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,n+1-a]}; NestList[nxt,{1,1},90][[;;,2]] (* Harvey P. Dale, Aug 31 2023 *)

q=vector(100); q[1]=1; for(n=2, #q, q[n] = if(q[n-1]%2, n-q[n-1], q[n-1]/2)); q

cp's OEIS Frontend

A185038 Positions at which A090895 (a(1) = 1 then a(n) = a(n-1)/2 if a(n-1) is even, a(n) = a(n-1) + n otherwise) reach the value of 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Extensions

A208852 Smallest m such that A090895(m) = n.

Views

Author

Keywords

Comments

Links

Programs

Haskell

A135287 a(0)=1; for n > 0, a(n) = a(n-1)+n if a(n-1) is odd, else a(n) = a(n-1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A208884 a(n) = (a(n-1) + n)/2^k where 2^k is the largest power of 2 dividing a(n-1) + n, for n>1 with a(1)=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI