A035524 -id:A035524 - cp's OEIS frontend

A030103 Base 4 reversal of n (written in base 10).

0, 1, 2, 3, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 1, 17, 33, 49, 5, 21, 37, 53, 9, 25, 41, 57, 13, 29, 45, 61, 2, 18, 34, 50, 6, 22, 38, 54, 10, 26, 42, 58, 14, 30, 46, 62, 3, 19, 35, 51, 7, 23, 39, 55, 11, 27, 43, 59, 15, 31, 47, 63, 1, 65, 129, 193, 17, 81, 145, 209, 33, 97, 161

Offset: 0

David W. Wilson

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

Cf. A004086, A030101 - A030108, A048703, A055948, A035524.

import Data.List (unfoldr)
a030103 n = foldl (\v d -> 4*v + d) 0 $ unfoldr dig n where
    dig x = if x == 0 then Nothing else Just $ swap $ divMod x 4
-- Reinhard Zumkeller, Oct 10 2011

IntegerReverse[Range[0, 100], 4] (* Paolo Xausa, Aug 07 2024 *)

a(n,b=4)=subst(Polrev(base(n,b)),x,b) /* where */
base(n,b)={my(a=[n%b]);while(0M. F. Hasler, Nov 04 2011
(MIT/GNU Scheme)
(define (A030103 n) (if (zero? n) n (let ((uplim (+ (A000523 n) (- 1 (modulo (A000523 n) 2))))) (add (lambda (i) (* (bit_i n (+ i (expt -1 i))) (expt 2 (- uplim i)))) 0 uplim))))
(define (bit_i n i) (modulo (floor->exact (/ n (expt 2 i))) 2))
;; The functional add implements sum_{i=lowlim..uplim} intfun(i):
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Antti Karttunen, Oct 30 2013

A035522 Reverse and add (in binary) - written in base 10.

Original entry on oeis.org

1, 2, 3, 6, 9, 18, 27, 54, 81, 150, 255, 510, 765, 1530, 2295, 6120, 6885, 12240, 13005, 24480, 25245, 48960, 49725, 97920, 98685, 196224, 197757, 392448, 393981, 785664, 788733, 1571328, 1574397, 3144192, 3150333, 6288384

Offset: 0

N. J. A. Sloane, E. M. Rains

			6 = 110, 110 + 011 = 1001 = 9, so 6 is followed by 9.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A035526, A035524.

a035522 n = a035522_list !! n
a035522_list = iterate a055944 1
-- Reinhard Zumkeller, Oct 21 2011

nxt[n_]:=Module[{idn2=IntegerDigits[n,2]},FromDigits[idn2+ Reverse[ idn2],2]]; NestList[nxt,1,40] (* Harvey P. Dale, Oct 02 2011 *)

a(n+1) = A055944(a(n)), a(0) = 1. [Reinhard Zumkeller, Nov 14 2011]

A055948 n + reversal of base 4 digits of n (written in base 10).

Original entry on oeis.org

0, 2, 4, 6, 5, 10, 15, 20, 10, 15, 20, 25, 15, 20, 25, 30, 17, 34, 51, 68, 25, 42, 59, 76, 33, 50, 67, 84, 41, 58, 75, 92, 34, 51, 68, 85, 42, 59, 76, 93, 50, 67, 84, 101, 58, 75, 92, 109, 51, 68, 85, 102, 59, 76, 93, 110, 67, 84, 101, 118, 75, 92, 109, 126, 65, 130, 195

Offset: 0

Henry Bottomley, Jul 18 2000

If n has an even number of digits in base 4 then a(n) is a multiple of 5.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Reverse and Add!

Cf. A030103, A056964.

Cf. A035524 (iterated).

a055948 n = n + a030103 n
-- Reinhard Zumkeller, Oct 10 2011

Table[n+FromDigits[Reverse[IntegerDigits[n,4]],4],{n,0,70}] (* Harvey P. Dale, Nov 24 2021 *)

a(n) = n + A030103(n).

A091675 Positive integers n such that the trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not join the trajectory of any m < n.

Original entry on oeis.org

1, 3, 21, 22, 26, 29, 31, 55, 256, 258, 262, 266, 269, 271, 282, 286, 287, 302, 312, 413, 479, 511, 519, 551, 555, 719, 795, 799, 1026, 1029, 1034, 1037, 1066, 1549, 1790, 2863, 3087, 3119, 4096, 4098, 4102, 4104, 4106, 4108, 4109, 4113, 4114, 4116, 4117

Offset: 1

Klaus Brockhaus, Jan 28 2004

The conjecture that the base-4 trajectories of the terms do not join is based on the observation that if the trajectories of two integers below 4120 join, this happens after at most 28 steps, while for any two terms listed above the trajectories do not join within 1000 steps. For pairs from 1, 3, 21, 22, 26, 29, 31, 55 this has even been checked for 5000 steps.

Base-4 analog of A070788.

			The trajectory of 2 is part of the trajectory of 1 (cf. A035524); the trajectory of 3 does not join the trajectory of 1 within 10000 steps; the trajectory of 21 does not join the trajectory of 1 or of 3 within 10000 steps.

Klaus Brockhaus, Illustration: Distribution of terms below 2000000
Index entries for sequences related to Reverse and Add!

Cf. A035524, A075421, A070788.

limit = 10^3; utraj = {};
Select[Range[4120], (x = NestList[ # + IntegerReverse[#, 4] &, #, limit]; If[Intersection[x, utraj] == {}, utraj = Union[utraj, x]; True, utraj = Union[utraj, x]]) &] (* Robert Price, Oct 20 2019 *)

A091678 In base 4, numbers n of the form k + reverse(k) for at least one k.

Original entry on oeis.org

0, 2, 4, 5, 6, 10, 15, 17, 20, 25, 30, 33, 34, 41, 42, 50, 51, 58, 59, 65, 67, 68, 75, 76, 84, 85, 92, 93, 101, 102, 105, 109, 110, 118, 125, 126, 130, 145, 150, 165, 170, 185, 190, 195, 210, 215, 230, 235, 250, 255, 257, 260, 275, 280, 289, 295, 300, 315, 320, 321

Offset: 0

Klaus Brockhaus, Jan 28 2004

Base-4 analog of A067030. Complement of A091679.

			25 is a term since 25 (decimal) = 121 = 110 + 011 = 20 (decimal) + 5 (decimal).

Cf. A035524, A091679, A067030.

A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

Original entry on oeis.org

290, 78, 18, 6, 3, 36, 21, 19, 7, 8, 4, 2, 1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 0

Klaus Brockhaus, Jan 28 2004

Conjecture 1: For each k > 0 the trajectory of k eventually leads to a term in the trajectory of some j which belongs to A075421, i.e., whose trajectory (presumably) never leads to a palindrome. Conjecture 2: There is no k > 0 such that the trajectory of k contains more than twelve palindromes, i.e., a(n) = -1 for n > 12.

Base-4 analog of A077594.

			a(4) = 3 since the trajectory of 3 contains the four palindromes 3, 15, 975, 64575 (3, 33, 33033, 3330333 in base 4) and at 20966400 joins the trajectory of 318 = A075421(2) and the trajectories of 1 (A035524) and 2 do not contain exactly four palindromes.

Index entries for sequences related to Reverse and Add!

Cf. A075299, A035524, A014192, A075420, A075421, A077594.

cp's OEIS Frontend

A030103 Base 4 reversal of n (written in base 10).

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A035522 Reverse and add (in binary) - written in base 10.

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A055948 n + reversal of base 4 digits of n (written in base 10).

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A091675 Positive integers n such that the trajectory of n under the Reverse and Add! operation carried out in base 4 (presumably) does not join the trajectory of any m < n.

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A091678 In base 4, numbers n of the form k + reverse(k) for at least one k.

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A091680 Smallest number whose base-4 Reverse and Add! trajectory (presumably) contains exactly n base-4 palindromes, or -1 if there is no such number.

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