A230638 -id:A230638 - cp's OEIS frontend

A230637 Leading power of 4 in A230638.

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

A053737 Sum of digits of (n written in base 4).

Original entry on oeis.org

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

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3}, 1->{1,2,3,4}, 2->{2,3,4,5}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1+1+0 = 2 because 20 is written as 110 base 4.
From _Omar E. Pol_, Feb 21 2010: (Start)
This can be written as a triangle (cf. A000120):
  0,
  1,2,3,
  1,2,3,4,2,3,4,5,3,4,5,6,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,5,6,7,5,6,7,8,6,7,8,9,
  1,2,3,4,2,3,4,5,3,4,5,6,4,5,6,7,2,3,4,5,3,4,5,6,4,5,6,7,5,6,7,8,3,4,5,6,4,...
where the rows converge to A173524.
(End)

Donovan Johnson, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS Vol. 12 (2009), Article 09.5.6.
Steve Butler and Ron L. Graham, Shuffling with ordered cards, arXiv 1003:4422 [math.CO], 2010.
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A231664-A231667, A138530.
Cf. A173524. - Omar E. Pol, Feb 21 2010
Sum of digits of n written in bases 2-16: A000120, A053735, this sequence, A053824, A053827, A053828, A053829, A053830, A007953, A053831, A053832, A053833, A053834, A053835, A053836.
Related base-4 sequences: A053737, A230631, A230632, A010064, A230633, A230634, A230635, A230636, A230637, A230638, A010065 (trajectory of 1).
Cf. A007090, A239690.

a053737 n = if n == 0 then 0 else a053737 m + r where (m, r) = divMod n 4
-- Reinhard Zumkeller, Mar 19 2015

for u=0:104; sol(u+1)=sum(dec2base(u,4)-'0');end
sol % Marius A. Burtea, Jan 17 2019

[&+Intseq(n,4):n in [0..104]]; // Marius A. Burtea, Jan 17 2019

A053737 := proc(n)
    add(d,d=convert(n,base,4)) ;
end proc: # R. J. Mathar, Oct 31 2012

Table[Plus @@ IntegerDigits[n, 4], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> {a, a+1, a+2, a+3}] &, {0}, 4] (* Robert G. Wilson v, Jul 27 2006 *)
DigitSum[Range[0, 100], 4] (* Paolo Xausa, Aug 01 2024 *)

PARI
```
a(n)=if(n<1,0,if(n%4,a(n-1)+1,a(n/4)))
```

a(n) = sumdigits(n, 4); \\ Michel Marcus, Aug 24 2019

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(4n+i) = a(n)+i for 0 <= i <= 3.
a(n) = n - 3*Sum_{k>0} floor(n/4^k) = n - 3*A054893(n). (End)
G.f.: (Sum_{k>=0} (x^(4^k) + 2*x^(2*4^k) + 3*x^(3*4^k))/(1 + x^(4^k) + x^(2*4^k) + x^(3*4^k)))/(1-x). - Franklin T. Adams-Watters, Nov 03 2005
a(n) = A138530(n,4) for n > 3. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A030386(n,k). - Philippe Deléham, Oct 21 2011
a(n) = A007953(A007090(n)). - Reinhard Zumkeller, Mar 19 2015
a(0) = 0; a(n) = a(n - 4^floor(log_4(n))) + 1. - Ilya Gutkovskiy, Aug 23 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 4*log(4)/3 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A010064 Base 4 self or Colombian numbers (not of form k + sum of base 4 digits of k).

Original entry on oeis.org

1, 3, 8, 13, 18, 20, 25, 30, 35, 37, 42, 47, 52, 54, 59, 64, 73, 78, 83, 85, 90, 95, 100, 102, 107, 112, 117, 119, 124, 129, 138, 143, 148, 150, 155, 160, 165, 167, 172, 177, 182, 184, 189, 194, 203, 208, 213, 215, 220, 225, 230, 232, 237, 242, 247, 249, 254

Offset: 1

Leonid Broukhis

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.24, pp. 179-180.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 384-386.

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

Cf. A003052, A010061, A010067, A010070, A230631-A230635.

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

s[n_] := n + Plus @@ IntegerDigits[n, 4]; m = 250; Complement[Range[m], Array[s, m]] (* Amiram Eldar, Nov 28 2020 *)

A230640 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives M(n).

Original entry on oeis.org

0, 4, 28, 248, 129140168, 68630377364912, 2088595827392656793085408064780643444068898148936888424953199350296

Offset: 1

N. J. A. Sloane, Oct 31 2013

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.

Cf. A230639.
Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1)
Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0, 0]);
else
B:=(f(ceil(n/2))[2] + f(floor(n/2))[2] + 2)/2;
M:=3^B+f(floor(n/2))[2]+1; RETURN([B, M]); fi;
end proc;
[seq(f(n)[2], n=1..7)];

A006064 Smallest junction number with n generators.

Original entry on oeis.org

0, 101, 10000000000001, 1000000000000000000000102

Offset: 1

N. J. A. Sloane

Strictly speaking, a junction number is a number n with more than one solution to x+digitsum(x) = n. However, it seems best to start this sequence with n=0, for which there is just one solution, x=0. - N. J. A. Sloane, Oct 31 2013.
a(3) = 10^13 + 1 was found by Narasinga Rao, who reports that Kaprekar verified that it is the smallest term. No details of Kaprekar's proof were given.
a(4) = 10^24 + 102 was conjectured by Narasinga Rao.
a(5) = 10^1111111111124 + 102. - Conjectured by Narasinga Rao, confirmed by Max Alekseyev and N. J. A. Sloane.
a(6) = 10^2222222222224 + 10000000000002. - Max Alekseyev
a(7) = 10^( (10^24 + 10^13 + 115) / 9 ) + 10^13 + 2. - Max Alekseyev
a(8) = 10^( (2*10^24 + 214)/9 ) + 10^24 + 103. - Max Alekseyev

			a(2) = 101 since 101 is the smallest number with two generators: 101 = A062028(91) = A062028(100).
a(4) = 10^24 + 102 = 1000000000000000000000102 has exactly four inverses w.r.t. A062028, namely 999999999999999999999893, 999999999999999999999902, 1000000000000000000000091 and 1000000000000000000000100.

M. Gardner, Time Travel and Other Mathematical Bewilderments. Freeman, NY, 1988, p. 116.
D. R. Kaprekar, The Mathematics of the New Self Numbers, Privately printed, 311 Devlali Camp, Devlali, India, 1963.
Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Max Alekseyev, Table of expressions for a(n), for n=1..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.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
D. R. Kaprekar, The Mathematics of the New Self Numbers [annotated and scanned]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.
Terry Trotter, Charlene numbers
Index entries for Colombian or self numbers and related sequences

Cf. A003052, A230093, A230100, A230303, A230857 (highest power of 10).

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

a(n) = the smallest m such that there are exactly n solutions to A062028(x)=m.

Edited, a(5)-a(6) added by Max Alekseyev, Jun 01 2011

a(1) added, a(5) corrected, a(7)-a(8) added by Max Alekseyev, Oct 26 2013

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

Original entry on oeis.org

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

A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

Original entry on oeis.org

0, 5, 129, 4102, 87112285931760246646623899502532662132742, 1852673427797059126777135760139006525652319754650249024631321344126610074239106

Offset: 1

N. J. A. Sloane, Oct 24 2013; Mar 26 2014

nonn

M(n) is the smallest value of k such that A228085(k) = n. For example, 129 is the first time a 3 appears in A228085 (and is therefore the first term in A230092). M(4) = 4102 is the first time a 4 appears in A228085 (and is therefore the first term in A227915).

			The terms are a(1) = 0, a(2) = 2^2+0+1, a(3) = 2^7+0+1, a(4) = 2^12+5+1, a(5) = 2^136+5+1, a(6) = 2^160+129+1, a(7) = 2^4233+129+1, a(8) = 2^8206+4102+1, a(9) = 2^k+4102+1 with k=2^136+4110, ... .
The length (in bits) of the n-th term is A230302(n)+1.

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. A228085, A230092, A227915, A230093, A230302 (for B(n)), A230864.

Smallest number m such that u + (sum of base-b digits of u) = m has exactly n solutions, for bases 2 through 10: A230303, A230640, A230638, A230867, A238840, A238841, A238842, A238843, A006064.

f:=proc(n) option remember; local B, M;
if n<=1 then RETURN([0,0]);
else
if (n mod 2) = 0 then B:=2*f(n/2)[2]+2;
else B:=f((n+1)/2)[2]+f((n-1)/2)[2]+2; fi;
M:=2^B+f(floor(n/2))[2]+1; RETURN([B,M]); fi;
end proc;
[seq(f(n)[2],n=1..6)];

Define i by 2^(i-1) < n <= 2^i. Then it appears that
a(n) = 2^2^2^...^2^x
a tower of height i+3, containing i+2 2's, where x is in the range 0 < x <= 1.
For example, if n=7, i=3, and
a(7) = 2^4233+130 = 2^2^2^2^2^.88303276...
Note also that i+2 = A230864(a(n)).

a(1)-a(8) were found by Donovan Johnson, Oct 22 2013.

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)

cp's OEIS Frontend

A230637 Leading power of 4 in A230638.

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A053737 Sum of digits of (n written in base 4).

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A010064 Base 4 self or Colombian numbers (not of form k + sum of base 4 digits of k).

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A230640 Let M(1)=0 and for n>1, B(n)=(M(ceiling(n/2))+M(floor(n/2))+2)/2, M(n)=3^B(n)+M(floor(n/2))+1. This sequence gives M(n).

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A006064 Smallest junction number with n generators.

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

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A230303 Let M(1)=0 and for n >= 2, let B(n)=M(ceiling(n/2))+M(floor(n/2))+2, M(n)=2^B(n)+M(floor(n/2))+1; sequence gives M(n).

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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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