A053737 -id:A053737 - cp's OEIS frontend

A010065 a(n+1) = a(n) + sum of digits in base 4 representation of a(n), with a(0) = 1.

1, 2, 4, 5, 7, 11, 16, 17, 19, 23, 28, 32, 34, 38, 43, 50, 55, 62, 70, 74, 79, 86, 91, 98, 103, 110, 118, 125, 133, 137, 142, 149, 154, 161, 166, 173, 181, 188, 196, 200, 205, 212, 217, 224, 229, 236, 244, 251, 262, 266, 271, 278, 283, 290, 295

Offset: 0

Leonid Broukhis

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24.

Donovan Johnson, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A010061, A010062, A010064, A004207.

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

a010065 n = a010065_list !! n
a010065_list = iterate a230631 1  -- Reinhard Zumkeller, Mar 20 2015

a(n+1) = A230631(a(n)). - Reinhard Zumkeller, Mar 20 2015

More terms from Neven Juric, Apr 11 2008

A027615 Number of 1's when n is written in base -2.

Original entry on oeis.org

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

Offset: 0

Pontus von Brömssen, Nov 14 1997

Base -2 is also called "negabinary".
From Jianing Song, Oct 18 2018: (Start)
Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(n), n >= 0. See A320642 for the other half of f.
For k > 0, the earliest occurrence of k is n = A305750(k).
Conjecture: a(n) != A053737(n) if and only if there exists even k >= 4 such that n mod 2^k >= (5*2^(k+1) + 2)/3. If this holds, then the probability of a random chosen number n to satisfy a(n) != A053737(n) is 1/6. (End)

			A039724(7) = 11011 which has four 1's, so a(7) = 4.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 164.

Jianing Song, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Negabinary.
Wikipedia, Negative base.

Cf. A000120, A005351, A039724, A053737, A072894, A305750, A320642.

a[n_] := a[n] = a[Quotient[n - 1, -2]] + Mod[n, 2]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Jul 23 2023 *)

a(n) = if(n==0, 0, a(n\(-2))+n%2) /* Jianing Song, Oct 18 2018 */

a(n) = 3*A072894(n+1) - 2*n - 3. Proof by Nikolaus Meyberg, following a conjecture by Ralf Stephan. - R. J. Mathar, Jan 11 2013
a(n) == n (mod 3). - Jianing Song, Oct 18 2018
a(n) = A000120(A005351(n)). - Michel Marcus, Oct 23 2018

A115362 Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3

Offset: 0

Paul Barry, Jan 21 2006

Row sums of the matrix product A115358*A115361.
Generalized Ruler Function for k=4. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
a(n) is 1 + the 4-adic valuation of n+1. - Joerg Arndt, Oct 07 2015

Antti Karttunen, Table of n, a(n) for n = 0..16384
Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=4.)

Cf. A053737, A115358, A115361, quadrisection of A235127.

a[ n_] := If[ n < 0, 0, 1 + IntegerExponent[n + 1, 4]]; (* Michael Somos, Jul 19 2017 *)

a(n) = 1 + valuation(n+1,4); \\ Joerg Arndt, Oct 07 2015

{a(n) = if( n<0, 0, n%4==3, 1 + a((n - 3) / 4), 1)}; /* Michael Somos, Jul 13 2017 */

[(1/3)*(4-sum(n.digits(4))+sum((n-1).digits(4))) for n in [1..96]] # Tom Edgar, Oct 06 2015

G.f.: Sum_{k>=0} x^(4^k)/(1-x^(4^k)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)
Dirichlet g.f. (conjectured): zeta(s)/(1-2^(-2s)). - Ralf Stephan, Mar 27 2015
a(n) = (1/3)*(4 + A053737(n) - A053737(n+1)). - Tom Edgar, Oct 06 2015
a(4*n) = a(4*n+1) = a(4*n+2) = 1, a(4*n+3) = 1+a(n), if n >= 0. - Michael Somos, Jul 13 2017
a(n) = 1 + A235127(1+n). - Antti Karttunen, Nov 18 2017, after Joerg Arndt's Oct 07 2015 comment.

A230632 Number of integers m such that m + (sum of digits in base-4 representation of m) = n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 2, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 0, 2, 0

Offset: 0

N. J. A. Sloane, Oct 30 2013

Number of occurrences of n in A230631.

Donovan Johnson, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A230631, A010064 (positions of 0's), A230633-A230635.

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634,A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A230631 a(n) = n + (sum of digits in base-4 representation of n).

Original entry on oeis.org

0, 2, 4, 6, 5, 7, 9, 11, 10, 12, 14, 16, 15, 17, 19, 21, 17, 19, 21, 23, 22, 24, 26, 28, 27, 29, 31, 33, 32, 34, 36, 38, 34, 36, 38, 40, 39, 41, 43, 45, 44, 46, 48, 50, 49, 51, 53, 55, 51, 53, 55, 57, 56, 58, 60, 62, 61, 63, 65, 67, 66, 68, 70, 72, 65, 67, 69, 71, 70, 72, 74, 76, 75, 77, 79, 81, 80, 82, 84, 86, 82

Offset: 0

N. J. A. Sloane, Oct 30 2013

Donovan Johnson, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences

Cf. A010064 (missing numbers), A230632 (number of inverses), A230633-A230635.

Related base-4 sequences: A053737, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

a230631 n = a053737 n + n  -- Reinhard Zumkeller, Mar 20 2015

apply( {A230631(n)=n+sumdigits(n,4)}, [0..99]) \\ M. F. Hasler, Oct 30 2024

a(n) = A053737(n) + n. - Reinhard Zumkeller, Mar 20 2015

A230633 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly one solution.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 9, 10, 11, 12, 14, 15, 16, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 56, 57, 58, 60, 61, 62, 63, 66, 68, 69, 71, 74, 75, 76, 77, 79, 80, 81, 87, 88, 89, 91, 92, 93, 94, 96, 97, 98, 104, 105, 106, 108, 109, 110, 111, 113, 114, 115, 121, 122, 123, 125, 126, 127

Offset: 1

N. J. A. Sloane, Oct 30 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Cf. A010064, A230631-A230635.

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A230634 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly two solutions.

Original entry on oeis.org

17, 19, 21, 34, 36, 38, 51, 53, 55, 65, 67, 70, 72, 82, 84, 86, 99, 101, 103, 116, 118, 120, 130, 132, 135, 137, 147, 149, 151, 164, 166, 168, 181, 183, 185, 195, 197, 200, 202, 212, 214, 216, 229, 231, 233, 246, 248, 250, 257, 261, 262, 263, 267, 274, 276, 278, 291, 293, 295, 308, 310, 312, 322, 324, 327, 329, 339

Offset: 1

N. J. A. Sloane, Oct 30 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

etsQ[n_]:=Count[#+Total[IntegerDigits[#,4]]&/@Range[n-1],n]==2; Select[ Range[ 350],etsQ] (* Harvey P. Dale, May 25 2016 *)

A230635 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly three solutions.

Original entry on oeis.org

16385, 16387, 16402, 16404, 32770, 32772, 32787, 32789, 49155, 49157, 49172, 49174, 65542, 65554, 81922, 81924, 81939, 81941, 98307, 98309, 98324, 98326, 114692, 114694, 114709, 114711, 131079, 131091, 147459, 147461, 147476, 147478, 163844, 163846, 163861

Offset: 1

N. J. A. Sloane, Oct 30 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A230636 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly four solutions.

Original entry on oeis.org

16777234, 33554451, 50331668, 83886099, 100663316, 117440533, 150994964, 167772181, 184549398, 218103829, 234881046, 251658263, 268435476, 268435478, 285212691, 301989908, 318767125, 352321556, 369098773, 385875990, 419430421, 436207638, 452984855, 486539286

Offset: 1

Donovan Johnson and N. J. A. Sloane, Oct 31 2013

Donovan Johnson, Table of n, a(n) for n = 1..10000
Index entries for Colombian or self numbers and related sequences

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

A230637 Leading power of 4 in A230638.

Original entry on oeis.org

2, 7, 12, 5468, 10924, 5597880, 16777229

Offset: 2

N. J. A. Sloane, Oct 31 2013

a(9) = ( 4^5468 + 2*4^12 + 39 ) / 3.
a(10) = 4^5468 + 13.
a(11) = ( 4^10924 + 2*4^5468 + 16407 ) / 3.
a(12) = 4^10924 + 10925
a(13) = ( 4^5597880 + 3*4^10924 + 32793 ) / 3.
a(14) = ( 2*4^5597880 + 32812 ) / 3.
a(15) = ( 4^16777229 + 4^5597880 + 2*4^12 + 16427 ) / 3.
a(16) = ( 2*4^16777229 + 4^13 + 42 ) / 3.

Max Alekseyev, Table n, expression for a(n) for n=2..100
Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; Journal of Combinatorics and Number Theory 12:3 (2022), 115-155.
Index entries for Colombian or self numbers and related sequences

Cf. A230638.

Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1)

Terms a(8) onward from Max Alekseyev, Oct 31 2013

cp's OEIS Frontend

A010065 a(n+1) = a(n) + sum of digits in base 4 representation of a(n), with a(0) = 1.

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A027615 Number of 1's when n is written in base -2.

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A115362 Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).

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A230632 Number of integers m such that m + (sum of digits in base-4 representation of m) = n.

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A230631 a(n) = n + (sum of digits in base-4 representation of n).

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A230633 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly one solution.

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A230634 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly two solutions.

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A230635 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly three solutions.

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A230636 Numbers n such that m + (sum of digits in base-4 representation of m) = n has exactly four solutions.

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