A007090 -id:A007090 - cp's OEIS frontend

A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

4, 12, 48, 192, 768, 3072, 12288, 49152, 196608, 786432, 3145728, 12582912, 50331648, 201326592, 805306368, 3221225472, 12884901888, 51539607552, 206158430208, 824633720832, 3298534883328, 13194139533312, 52776558133248

Offset: 1

Jonathan Vos Post, Jul 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4).

Cf. A000302, A007090, A081604, A110591, A110593.

Concatenation([4],List([2..25],n->3*4^(n-1))); # Muniru A Asiru, Oct 21 2018

[4] cat [3*4^(n-1): n in [2..30]]; // Vincenzo Librandi, May 29 2014

seq(coeff(series(4*x*(1-x)/(1-4*x),x,n+1), x, n), n = 1 .. 25); # Muniru A Asiru, Oct 21 2018

CoefficientList[Series[4 (1 - x)/(1 - 4 x), {x, 0, 40}], x] (* Vincenzo Librandi, May 29 2014 *)

x='x+O('x^50); Vec(4*x*(1 - x)/(1 - 4*x)) \\ G. C. Greubel, Sep 01 2017

a(n) = A002001(n), n>1. - R. J. Mathar, Aug 18 2008

G.f.: 4*x*(1 - x)/(1 - 4*x). - Vincenzo Librandi, May 29 2014

Definition corrected by R. J. Mathar, Aug 18 2008

A117640 Concatenation of first n numbers in base 4.

Original entry on oeis.org

1, 12, 123, 12310, 1231011, 123101112, 12310111213, 1231011121320, 123101112132021, 12310111213202122, 1231011121320212223, 123101112132021222330, 12310111213202122233031

Offset: 1

Jonathan Vos Post, Apr 27 2006

Concatenation of the first n terms of A007090.

Base-4 analog of A058935.

N. J. A. Sloane, Table of n, a(n) for n = 1..35

Cf. A030373, A048436.

Other bases: A058935 (2), A360502 (3), A007908 (10).

Table[FromDigits[Flatten[Table[IntegerDigits[n,4],{n,k}]]],{k,15}] (* Harvey P. Dale, Jan 18 2023 *)

from gmpy2 import digits
def A117640(n): return int(''.join(digits(n,4) for n in range(1,n+1))) # Chai Wah Wu, Apr 19 2023

Edited by Jason Kimberley, Nov 27 2012

A160382 Number of 2's in base-4 representation of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2

Offset: 0

Frank Ruskey, Jun 05 2009

Dominic McCarty, Table of n, a(n) for n = 0..10000
Franklin T. Adams-Watters and Frank Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.

Cf. A007090.

DigitCount[Range[0,110],4,2] (* Harvey P. Dale, Sep 09 2024 *)

a(n) = #select(x->(x==2), digits(n, 4)); \\ Michel Marcus, Mar 24 2020

Recurrence relation: a(0) = 0, a(4m+2) = 1+a(m), a(4m) = a(4m+1) = a(4m+3) = a(m).
G.f.: (1/(1-z))*Sum_{m>=0} (z^(2*4^m)*(1 - z^(4^m))/(1 - z^(4^(m+1)))).
Morphism: 0, j -> j,j,j+1,j; e.g., 0 -> 0010 -> 0010110111210010 -> ...

A239690 Base 4 sum of digits of prime(n).

Original entry on oeis.org

2, 3, 2, 4, 5, 4, 2, 4, 5, 5, 7, 4, 5, 7, 8, 5, 8, 7, 4, 5, 4, 7, 5, 5, 4, 5, 7, 8, 7, 5, 10, 5, 5, 7, 5, 7, 7, 7, 8, 8, 8, 7, 11, 4, 5, 7, 7, 10, 8, 7, 8, 11, 7, 11, 2, 5, 5, 7, 4, 5, 7, 5, 7, 8, 7, 8, 7, 4, 8, 7, 5, 8, 10, 7, 10, 11, 5, 7, 5, 7, 8, 7, 11, 7

Offset: 1

Tom Edgar, Mar 24 2014

a(n) is the rank of prime(n) in the base-4 dominance order on the natural numbers.

			The sixth prime is 13, 13 in base 4 is (3,1) so a(6)=3+1=4.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Tyler Ball and Daniel Juda, Dominance over N, Rose-Hulman Undergraduate Mathematics Journal, Vol. 13, No. 2, Fall 2013.

Cf. A007605, A014499, A053737, A239619.

Cf. A000040, A007090.

a239690 = a053737 . a000040  -- Reinhard Zumkeller, Mar 20 2015

[&+Intseq(NthPrime(n),4): n in [1..100]]; // Vincenzo Librandi, Mar 25 2014

Table[Plus @@ IntegerDigits[Prime[n], 4], {n, 1, 100}] (* Vincenzo Librandi, Mar 25 2014 *)

[sum(i.digits(base=4)) for i in primes_first_n(200)]

a(n) = A053737(A000040(n)).

A262101 Pseudoprimes to base 4, written in base 4.

Original entry on oeis.org

33, 1111, 1123, 11111, 12303, 13003, 20301, 22011, 22333, 101101, 103133, 103313, 111223, 120231, 122133, 123001, 131203, 131301, 133333, 200113, 212201, 222031, 230011, 300331, 303031, 310213, 321203, 333001, 1010101, 1010103, 1021021, 1022323, 1023323, 1111111, 1112233, 1213021, 1213303, 1330111, 2002001, 2010201, 2013313, 2023033, 2031211, 2032223

Offset: 1

Abdul Gaffar Khan, Sep 11 2015

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007090 (numbers in base 4), A020136 (pseudoprimes to base 4).

Cf. A258189, A262102, A262103, A262104, A262105, A262154.

BaseForm[Select[Range[4096], Not[PrimeQ[#]] && PowerMod[4, # - 1, #] == 1 &], 4]

lista(nn, b=4) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007090(A020136(n)).

A292371 A binary encoding of 1-digits in the base-4 representation of n.

Original entry on oeis.org

0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 4, 5, 4, 4, 6, 7, 6, 6, 4, 5, 4, 4, 4, 5, 4, 4, 0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 2, 3, 2, 2, 0, 1, 0, 0, 0, 1, 0, 0, 8, 9, 8, 8, 10, 11, 10, 10, 8, 9, 8, 8, 8, 9, 8, 8, 12, 13, 12, 12, 14, 15, 14, 14, 12, 13, 12, 12, 12, 13, 12, 12, 8, 9, 8, 8, 10, 11, 10, 10, 8, 9, 8, 8, 8, 9, 8, 8, 8

Offset: 0

Antti Karttunen, Sep 15 2017

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        1          1           1
   2        0          2           0
   3        0          3           0
   4        2         10          10
   5        3         11          11
   6        2         12          10
   7        2         13          10
   8        0         20           0
   9        1         21           1
  10        0         22           0
  11        0         23           0
  12        0         30           0
  13        1         31           1
  14        0         32           0
  15        0         33           0
  16        4        100         100
  17        5        101         101
  18        4        102         100

Antti Karttunen, Table of n, a(n) for n = 0..65536
Rémy Sigrist, Interactive scatterplot of (a(n), A292372(n), A292373(n)) for n=0..4^8-1 [provided your web browser supports the Plotly library, you should see icons on the top right corner of the page: if you choose "Orbital rotation", then you will be able to rotate the plot alongside three axes, the 3D plot here corresponds to a Sierpiński triangle-based pyramid]
Index entries for sequences related to binary expansion of n

Cf. A003188, A004198, A007088, A007090, A059905, A160381, A292272, A292370, A292372, A292373.

Cf. A289813 (analogous sequence for base 3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 1, 1, 0], 2], {n, 0, 112}] (* Michael De Vlieger, Sep 21 2017 *)

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==1 else '0' for i in k), 2)
print([a(n) for n in range(116)]) # Indranil Ghosh, Sep 21 2017

def A292371(n): return int(bin(n&~(n>>1))[:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059905(A292272(n)) = A059905(n AND A003188(n)), where AND is bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160381(n).

A293657 Base-4 circular primes that are not base-4 repunits.

Original entry on oeis.org

7, 13, 23, 29, 53, 383, 509, 863, 983, 1013

Offset: 1

Felix Fröhlich, Oct 28 2017

Conjecture: The sequence is finite, with 1013 being the last term (see A293142).
Written in base 4 (A007090), the terms are 13, 31, 113, 131, 311, 11333, 13331, 31133, 33113, 33311. - Antti Karttunen, Nov 26 2017
From Michael De Vlieger, Dec 30 2017: (Start)
The digits of primes in this sequence must be in the reduced residue system modulo 4, i.e., {1, 3}.
a(11), if it exists, must be larger than 4^21 = 4398046511104. (End)

			53 written in base 4 is 311. The base-4 numbers 311, 131, 113 written in base 10 are 53, 29, 23, respectively and all those numbers are prime, so 23, 29 and 53 are terms of the sequence.

Cf. A007090, A293142.

Cf. base-b nonrepunit circular primes: A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=10).

With[{b = 4}, Select[Array[Map[If[Union@ # == {1}, 0, FromDigits[#, b]] &, NestList[RotateLeft, #, Length@ # - 1]] &@ IntegerDigits[Prime@ #, b] &, 10^6, If[PrimeQ@ b, #, # + 1] &@ PrimePi@ b], AllTrue[#, PrimeQ] &][[All, 1]] ] (* Michael De Vlieger, Nov 26 2017 *)
With[{b = 4}, Select[Flatten@ Array[FromDigits[#, b] & /@ Most@ Rest@ Tuples[{1, 3}, #] &, 18, 2], Function[w, And[ AllTrue[ Array[ FromDigits[ RotateRight[w, #], b] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 4))!=vecmax(digits(p, 4)), if(is_circularprime(p, 4), print1(p, ", "))))

A353112 Base-4 representation of A000422(n).

Original entry on oeis.org

1, 111, 11001, 1003201, 31100301, 2133233301, 131030232301, 11032113332301, 322313212202301, 22032322210302301, 100022230001210102301, 123202121121121130102301, 232213121102313132210102301, 1032130122202213111001210102301

Offset: 1

Seiichi Manyama, Apr 24 2022

Cf. A000422, A353110, A353111, A353113, A353114, A353115, A353116, A353117.

Cf. A007090, A353104.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.reverse.join.to_i.to_s(k).to_i}
end
p A(4, 20)

a(n) = A007090(A000422(n)).

A004053 For m=2,3,..., write m in bases 2,3,..,m.

Original entry on oeis.org

10, 11, 10, 100, 11, 10, 101, 12, 11, 10, 110, 20, 12, 11, 10, 111, 21, 13, 12, 11, 10, 1000, 22, 20, 13, 12, 11, 10, 1001, 100, 21, 14, 13, 12, 11, 10, 1010, 101, 22, 20, 14, 13, 12, 11, 10, 1011, 102, 23, 21, 15, 14, 13, 12, 11, 10, 1100, 110, 30, 22, 20, 15, 14, 13, 12, 11, 10

Offset: 2

Johan Boye (johbo(AT)ida.liu.se)

			Triangle begins:
    10;
    11,  10;
   100,  11, 10;
   101,  12, 11, 10;
   110,  20, 12, 11, 10;
   111,  21, 13, 12, 11, 10;
  1000,  22, 20, 13, 12, 11, 10;
  1001, 100, 21, 14, 13, 12, 11, 10;
  ...

Alois P. Heinz, Rows n = 2..20, flattened (row 21 would require digits > 9)

Cf. A007088, A007090, A007091, A007092, A007093, A007094, A007095.

FromDigits/@Flatten[Table[IntegerDigits[m,b],{m,2,20},{b,2,m}],1] (* Harvey P. Dale, Dec 01 2024 *)

T(n, k) = fromdigits(digits(n, k), 10);
tabl(nn) = for (n=2, nn, for (b=2, n, print1(T(n, b), ", "))); \\ Michel Marcus, Aug 30 2019

A037387 Numbers k such that every base-4 digit of k is a base-5 digit of k.

Original entry on oeis.org

1, 2, 3, 5, 10, 15, 21, 28, 37, 38, 42, 58, 63, 76, 80, 86, 132, 136, 137, 138, 142, 152, 160, 167, 178, 183, 190, 191, 202, 204, 205, 210, 213, 214, 215, 217, 220, 221, 222, 223, 238, 240, 256, 257, 258, 261, 266, 276, 277, 278

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007090, A007091.

Cf. A037388, A037389, A037390, A037391, A037392.

import Data.List ((\\), nub)
a037387 n = a037387_list !! (n-1)
a037387_list = filter f [1..] where
   f x = null $ nub (ds 4 x) \\ nub (ds 5 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[300],SubsetQ[IntegerDigits[#,5],IntegerDigits[#,4]]&] (* Harvey P. Dale, Mar 27 2019 *)

cp's OEIS Frontend

A110594 a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).

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A117640 Concatenation of first n numbers in base 4.

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A160382 Number of 2's in base-4 representation of n.

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A239690 Base 4 sum of digits of prime(n).

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A262101 Pseudoprimes to base 4, written in base 4.

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A292371 A binary encoding of 1-digits in the base-4 representation of n.

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A293657 Base-4 circular primes that are not base-4 repunits.

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