A007090 -id:A007090 - cp's OEIS frontend

A276326 Numbers expressed in greedy A001563-base.

0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 40, 41, 100, 101, 102, 103, 110, 111, 112, 113, 120, 121, 122, 123, 130, 131, 132, 133, 140, 141, 200, 201, 202, 203, 210, 211, 212, 213, 220, 221, 222, 223, 230, 231, 232, 233, 240, 241, 300, 301, 302, 303, 310, 311, 312, 313, 320, 321, 322, 323, 330, 331, 332, 333, 340, 341, 400

Offset: 0

Antti Karttunen, Aug 30 2016

Terms A001563(1) = 1, A001563(2) = 4, A001563(3) = 18, ... give the base values for the digit positions from 1 onward. Digit places are filled by always trying to find the largest possible term of A001563 that still fits into the sum.

A130744(8) = 3225600 = 10*A001563(8) is the first number which yields an ambiguous representation when expressed in decimal, because in this base it is actually "A0000000" (where digit "A" stands for ten).

			To recover n from a(n) the digits in positions i = 1, 2, 3, ... (starting indexing from the least significant digit at right) are multiplied by A001563(i) and added together:
  ----------------
   n         a(n)
  ----------------
   0           0
   1           1
   2           2
   3           3
   4          10
   5          11
   6          12
   7          13
   8          20
   9          21
  10          22
  11          23
  12          30
  13          31
  14          32
  15          33
  16          40
  17          41 (as 4*A001563(2) + 1*A001563(1) = 17)
  18         100 (as 1*A001563(3) + 0*A001563(2) + 0*A001563(1) = 18)
and:
3225599 99111111 (as 3225599 = 9*b(8) + 9*b(7) + b(6) + b(5) + b(4) + b(3) + b(2) + b(1)), where b(n) = A001563(n).

Antti Karttunen, Table of n, a(n) for n = 0..4320

Cf. A001563, A130744, A258198, A276335.
Cf. A276327 (the least significant nonzero digit).
Cf. A276328 (the sum of digits).
Cf. A276333 (the most significant digit).
Cf. A276336 (a largest digit).
Cf. A276337 (number of nonzero digits).
Cf. A033312 (repunits).
Cf. A276091 (no digits larger than one).
Differs from A007090 for the first time at n=16 and from A055655 at n=18.
Cf. also A007623, A007961, A000433, A014418, A265747.

f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], (# #!) &[# - i]]], {i, 0, # - 1}] &@ NestWhile[# + 1 &, 0, (# #!) &[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[FromDigits@ f@ n, {n, 72}] (* Michael De Vlieger, Aug 31 2016 *)

(define (A276326 n) (let loop ((n n) (s 0)) (if (zero? n) s (let ((dig (A276333 n))) (if (> dig 9) (error "A276326: ambiguous representation of n, digit > 9 would be needed: " n dig) (loop (A276335 n) (+ s (* dig (expt 10 (- (A258198 n) 1))))))))))

A004678 Primes written in base 4.

Original entry on oeis.org

2, 3, 11, 13, 23, 31, 101, 103, 113, 131, 133, 211, 221, 223, 233, 311, 323, 331, 1003, 1013, 1021, 1033, 1103, 1121, 1201, 1211, 1213, 1223, 1231, 1301, 1333, 2003, 2021, 2023, 2111, 2113, 2131, 2203, 2213, 2231, 2303, 2311, 2333, 3001, 3011, 3013

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Analogs in other bases: A004676 (base 2), A001363 (base 3), A004679 (base 5), A004680 (base 6), A004681 (base 7), A004682 (base 8), A004683 (base 9), A000040 (base 10), A004684 (base 11).

Cf. A072805 (primes of form 4k+3 written in base 3).

[Seqint(Intseq(NthPrime(n), 4)): n in [1..60]]; // G. C. Greubel, Oct 12 2018

FromDigits/@IntegerDigits[Prime[Range[50]],4] (* Vincenzo Librandi, Sep 03 2016 *)

a(n)=subst(Pol(digits(prime(n),4)),'x,10) \\ Charles R Greathouse IV, Nov 06 2013

a(n) = A007090(A000040(n)). - Jonathan Vos Post, Sep 09 2006

More terms from Vincenzo Librandi, Sep 03 2016

A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, -4, -3, -2, -1, -8, -7, -6, -5, -12, -11, -10, -9, 16, 17, 18, 19, 12, 13, 14, 15, 8, 9, 10, 11, 4, 5, 6, 7, 32, 33, 34, 35, 28, 29, 30, 31, 24, 25, 26, 27, 20, 21, 22, 23, 48, 49, 50, 51, 44, 45, 46, 47, 40, 41, 42, 43, 36, 37, 38, 39, -64, -63, -62, -61, -68, -67, -66, -65, -72, -71, -70, -69

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 4 representation for n converted from base -4 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625 [math.CO], 2020-2021.

Cf. A007090, A007608, A053985, A065369, A073792, A073793, A073794, A073795, A073796 & A073835.

Cf. A066321, A320283.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 4]], {n, 1, 80}]; b

a(n) = subst(Pol(digits(n,4)), x, -4); \\ Michel Marcus, Jan 30 2019

a(4*k+m) = -4*a(k)+m for 0 <= m < 4. - Chai Wah Wu, Jan 16 2020

A010343 Base-4 Armstrong or narcissistic positive numbers.

Original entry on oeis.org

1, 2, 3, 130, 131, 203, 223, 313, 332, 1103, 3303

Offset: 1

N. J. A. Sloane

Eric Weisstein's World of Mathematics, Narcissistic Number
D. T. Winter, Table of Armstrong Numbers

Cf. A010344 (a(n) written in base 10).

In other bases: A010345 (base 5), A010347 (base 6), A010349 (base 7), A010351 (base 8), A010352 (base 9), A005188 (base 10).

[fromdigits(digits(n,4))|n<-A010344] \\ M. F. Hasler, Nov 18 2019

a(n) = A007090(A010344(n)). - M. F. Hasler, Nov 18 2019

Edited by Joseph Myers, Jun 28 2009

"Positive" added to definition. - N. J. A. Sloane, Nov 18 2019

A061819 Multiples of 3 containing only digits 0,1,2,3.

Original entry on oeis.org

0, 3, 12, 21, 30, 33, 102, 111, 120, 123, 132, 201, 210, 213, 222, 231, 300, 303, 312, 321, 330, 333, 1002, 1011, 1020, 1023, 1032, 1101, 1110, 1113, 1122, 1131, 1200, 1203, 1212, 1221, 1230, 1233, 1302, 1311, 1320, 1323, 1332, 2001, 2010, 2013, 2022, 2031

Offset: 1

Amarnath Murthy, May 28 2001

			132 is a term containing digits less than 4.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for 10-automatic sequences.

Cf. A061818.

Select[3*Range[0,750],Max[IntegerDigits[#]]<4&] (* Harvey P. Dale, Jul 10 2014 *)

a(n) = A007090(3n), i.e. can also be read as multiples of 3 (A008585) written in base 4.

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Henry Bottomley, May 30 2001

A085184 Sequence A085183 shown in base 4. Quaternary code for binary trees.

Original entry on oeis.org

0, 1, 2, 11, 12, 21, 22, 30, 111, 112, 121, 122, 130, 211, 212, 221, 222, 230, 301, 302, 310, 320, 1111, 1112, 1121, 1122, 1130, 1211, 1212, 1221, 1222, 1230, 1301, 1302, 1310, 1320, 2111, 2112, 2121, 2122, 2130, 2211, 2212, 2221, 2222, 2230, 2301, 2302

Offset: 1

Antti Karttunen, Jun 14 2003

This sequence gives two alternative ways to encode rooted plane binary trees (Stanley's interpretation 'c' = interpretation 'd' without the outermost edges):
A: scan each term from left to right and for each 0, add a leaf node to the tree (terminate a branch), for each 1, add a leftward leaning branch \, for each 2, add a rightward leaning branch / and for each 3, add a double-branch \/ and continue in left-to-right, depth-first fashion.
B: Like method A, but the roles of digits 1 and 2 are swapped. When one compares the generated trees to the "standard order" as specified in the illustrations for A014486, one obtains the permutation A074684/A074683 for the case A and A082356/A082355 for the case B.
If we assign the following weights for each digit: w(0) = -1, w(1) = w(2) = 0, w(3) = +1, then the sequence gives all base-4 numbers for which all the partial sums of digit weights (from the most significant to the least significant end) are nonnegative and the final sum is zero. The initial term 0 is considered to have no significant digits at all, so its total weight is zero also.

			For the first eleven terms the following binary trees are constructed with method A. With method B we would get their mirror images, although this doesn't hold in general (e.g. for terms like 301-320).
........................................................\......./......\...
.....................\......./.......\......./...........\......\....../...
..*......\....../.....\......\......./....../.....\/......\......\.....\...
..0......1......2.....11.....12.....21.....22.....30....111....112....121..

R. P. Stanley, Exercises on Catalan and Related Numbers

Cf. A085185. Number of terms with n significant digits is given by A000108(n+1).

a(n) = A007090(A085183(n)).

A107715 Primes having only {0,1,2,3} as digits.

Original entry on oeis.org

2, 3, 11, 13, 23, 31, 101, 103, 113, 131, 211, 223, 233, 311, 313, 331, 1013, 1021, 1031, 1033, 1103, 1123, 1201, 1213, 1223, 1231, 1301, 1303, 1321, 2003, 2011, 2111, 2113, 2131, 2203, 2213, 2221, 2311, 2333, 3001, 3011, 3023, 3121, 3203, 3221, 3301, 3313

Offset: 1

Rick L. Shepherd, May 22 2005

Jason Bard, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)

Subsequence of A036956.
Cf. A036953 (primes containing digits from set {0, 1, 2}).
Cf. A010051, A007090, A193890, A385776.

a107715 n = a107715_list !! (n-1)
a107715_list = filter ((== 1) . a010051) a007090_list
-- Reinhard Zumkeller, Aug 11 2011

Select[Prime[Range[500]],Max[IntegerDigits[#]]<4&] (* Harvey P. Dale, May 09 2012 *)
Select[FromDigits/@Tuples[{0,1,2,3},4],PrimeQ] (* Harvey P. Dale, Mar 06 2016 *)

from gmpy2 import digits
from sympy import isprime
[int(digits(n,4)) for n in range(1000) if isprime(int(digits(n,4)))] # Chai Wah Wu, Jul 31 2014

print(list(islice(primes_with("0123"), 41))) # uses function/imports in A385776. Jason Bard, Jul 18 2025

A136333 Numbers containing only digits coprime to 10 in their decimal representation.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 17, 19, 31, 33, 37, 39, 71, 73, 77, 79, 91, 93, 97, 99, 111, 113, 117, 119, 131, 133, 137, 139, 171, 173, 177, 179, 191, 193, 197, 199, 311, 313, 317, 319, 331, 333, 337, 339, 371, 373, 377, 379, 391, 393, 397, 399, 711, 713, 717, 719, 731

Offset: 1

Reinhard Zumkeller, Mar 26 2008

Numbers containing digits 1,3,7,9 only, or, numbers written in base 4 (cf. A007090) with digits mapped by: 0->1, 1->3, 2->7 and 3->9. - Reinhard Zumkeller, Jul 17 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Index entries for 10-automatic sequences.

Cf. A007090, A091633 (primes), A245193.

import Data.List (intersect)
a136333 n = a136333_list !! (n-1)
a136333_list = filter (null . intersect "024568" . show) [1..]
-- Reinhard Zumkeller, Jul 17 2014

fQ[n_] := Block[{s = {1, 3, 7, 9}}, Union[Join[s, IntegerDigits@ n]] == s]; Select[ Range@ 1000, fQ] (* or *)
depth = 3; FromDigits@# & /@ FlattenAt[ Table[ Tuples[{1, 3, 7, 9}, n], {n, depth}], {#} & /@ Range[depth]] (* Robert G. Wilson v, Jul 02 2014 *)

isok(m) = my(d=digits(m)); apply(x->gcd(x, 10), d) == vector(#d, k, 1); \\ Michel Marcus, Feb 25 2022

Sum_{n>=1} 1/a(n) = 2.395867871130444522329053889312125689319669370758630349552737883715872077555... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

A346688 Replace 4^k with (-1)^k in base-4 expansion of n.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jul 29 2021

If n has base-4 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

			54 = 312_4, 2 - 1 + 3 = 4, so a(54) = 4.

Cf. A007090, A053737, A055017, A065359, A065368, A346689, A346690, A346691.

nmax = 104; A[] = 0; Do[A[x] = x (1 + 2 x + 3 x^2)/(1 - x^4) - (1 + x + x^2 + x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 5 Sum[(-1)^k Floor[n/4^k], {k, 1, Floor[Log[4, n]]}], {n, 0, 104}]

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 4)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 29 2021

G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2) / (1 - x^4) - (1 + x + x^2 + x^3) * A(x^4).

a(n) = n + 5 * Sum_{k>=1} (-1)^k * floor(n/4^k).

A037576 Base-4 digits are, in order, the first n terms of the periodic sequence with initial period 1,3.

Original entry on oeis.org

1, 7, 29, 119, 477, 1911, 7645, 30583, 122333, 489335, 1957341, 7829367, 31317469, 125269879, 501079517, 2004318071, 8017272285, 32069089143, 128276356573, 513105426295, 2052421705181, 8209686820727, 32838747282909

Offset: 1

Clark Kimberling

Partial sums of A255465. - Klaus Purath, Mar 18 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Rule 190
Index entries for sequences related to cellular automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (4,1,-4).

Cf. A007090 (numbers in base 4), A037582 (decimal), A265688 (binary), A118111.

I:=[1, 7, 29]; [n le 3 select I[n] else 4*Self(n-1)+Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

CoefficientList[Series[(1+3*x)/((x-1)*(4*x-1)*(1+x)),{x,0,30}],x] (*or*) LinearRecurrence[{4,1,-4},{1,7,29},40] (* Vincenzo Librandi, Jun 22 2012 *)

my(x='x+O('x^99)); Vec(x*(1+3*x)/((1-x)*(1-4*x)*(1+x))) \\ Altug Alkan, Sep 21 2018

print([7*4**n//15 for n in range(1,30)]) # Karl V. Keller, Jr., Mar 09 2021

G.f.: x*(1+3*x)/((1-x)*(1-4*x)*(1+x)). - Vincenzo Librandi, Jun 22 2012
a(n) = 4*a(n-1) + a(n-2) - 4*a(n-3). - Vincenzo Librandi, Jun 22 2012
a(n) = (7*4^n + 3*(-1)^n - 10)/15. - Bruno Berselli, Jun 22 2012, corrected by Klaus Purath, Mar 18 2021.
a(n) = floor(7*4^n/15). - Karl V. Keller, Jr., Mar 09 2021
From Klaus Purath, Mar 18 2021: (Start)
a(n) = 16*a(n-2) - 3*(-1)^n + 10, assuming that a(0) = 0.
a(n) = 4*a(n-1) + 2 + (-1)^n.
a(n) = 5*a(n-1) - 4*a(n-2) + 2*(-1)^n, n > 2. (End)

cp's OEIS Frontend

A276326 Numbers expressed in greedy A001563-base.

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A004678 Primes written in base 4.

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A073791 Replace 4^k with (-4)^k in base 4 expansion of n.

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A010343 Base-4 Armstrong or narcissistic positive numbers.

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A061819 Multiples of 3 containing only digits 0,1,2,3.

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A085184 Sequence A085183 shown in base 4. Quaternary code for binary trees.

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A107715 Primes having only {0,1,2,3} as digits.

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A136333 Numbers containing only digits coprime to 10 in their decimal representation.

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