cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A000695 Moser-de Bruijn sequence: sums of distinct powers of 4.

Original entry on oeis.org

0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, 85, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 1024, 1025, 1028, 1029, 1040, 1041, 1044, 1045, 1088, 1089, 1092, 1093, 1104, 1105, 1108, 1109, 1280, 1281, 1284, 1285
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although this is a list, it has offset 0 for both historical and mathematical reasons.
Numbers whose set of base-4 digits is a subset of {0,1}. - Ray Chandler, Aug 03 2004, corrected by M. F. Hasler, Oct 16 2018
Numbers k such that the sum of the base-2 digits of k = sum of the base-4 digits of k. - Clark Kimberling
Numbers having the same representation in both binary and negabinary (A039724). - Eric W. Weisstein
This sequence has many other interesting and useful properties. Every term k corresponds to a unique pair i,j with k = a(i) + 2*a(j) (i=A059905(n), j=A059906(n)) -- see A126684. Every list of numbers L = [L1,L2,L3,...] can be encoded uniquely by "recursive binary interleaving", where f(L) = a(L1) + 2*a(f([L2,L3,...])) with f([])=0. - Marc LeBrun, Feb 07 2001
This may be described concisely using the "rebase" notation b[n]q, which means "replace b with q in the expansion of n", thus "rebasing" n from base b into base q. The present sequence is 2[n]4. Many interesting operations (e.g., 10[n](1/10) = digit reverse, shifted) are nicely expressible this way. Note that q[n]b is (roughly) inverse to b[n]q. It's also natural to generalize the idea of "basis" so as to cover the likes of F[n]2, the so-called "fibbinary" numbers (A003714) and provide standard ready-made images of entities obeying other arithmetics, say like GF2[n]2 (e.g., primes = A014580, squares = the present sequence, etc.). - Marc LeBrun, Mar 24 2005
a(n) is also equal to the product n X n formed using carryless binary multiplication (A059729, A063010). - Henry Bottomley, Jul 03 2001
Numbers k such that A004117(k) is odd. - Pontus von Brömssen, Nov 25 2008
Fixed point of the morphism: 0 -> 01; 1 -> 45; 2 -> 89; ...; n -> (4n)(4n+1), starting from a(0)=0. - Philippe Deléham, Oct 22 2011
If n is even and present, so is n+1. - Robert G. Wilson v, Oct 24 2014
Also: interleave binary digits of n with 0's. (Equivalent to the "rebase" interpretation above.) - M. F. Hasler, Oct 16 2018
Named after the Austrian-Canadian mathematician Leo Moser (1921-1970) and the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - Amiram Eldar, Jun 19 2021
Conjecture: The sums of distinct powers of k > 2 can be constructed as the following (k-1)-ary rooted tree. For each n the tree grows and a(n) is then the total number of nodes. For n = 1, the root of the tree is added. For n > 1, if n is odd one leaf of depth n-2 grows one child. If n is even all leaves of depth >= (n - 1 - A000225(A001511(n/2))) grow the maximum number of children. An illustration is provided in the links. - John Tyler Rascoe, Oct 09 2022

Examples

			G.f.: x + 4*x^2 + 5*x^3 + 16*x^4 + 17*x^5 + 20*x^6 + 21*x^7 + 64*x^8 + ...
If n=27, then b_0=1, b_1=1, b_2=0, b_3=1, b_4=1. Therefore a(27) = 4^4 + 4^3 + 4 + 1 = 325; k = b_0 + b_2*2 + b_4*2^2 = 5, l = b_1 + b_3*2 = 3, such that a(5)=17, a(3)=5 and 27 = 17 + 2*5. - _Vladimir Shevelev_, Nov 10 2008

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For generating functions Product_{k>=0} (1 + a*x^(b^k)) for the following values of (a,b) see: (1,2) A000012 and A000027, (1,3) A039966 and A005836, (1,4) A151666 and A000695, (1,5) A151667 and A033042, (2,2) A001316, (2,3) A151668, (2,4) A151669, (2,5) A151670, (3,2) A048883, (3,3) A117940, (3,4) A151665, (3,5) A151671, (4,2) A102376, (4,3) A151672, (4,4) A151673, (4,5) A151674.
Main diagonal of A048720, second column of A048723.
Cf. A000225, A000302, A001511, A007583, A059884, A059901, A059904, A059905, A059906, A007088, A033042-A033052, A126684, A145812.
A062880(n) = 2*a(n); A001196(n) = 3*a(n).
Row 4 of array A104257.

Programs

  • C
    uint32_t a_next(uint32_t a_n) { return (a_n + 0xaaaaaaab) & 0x55555555; } /* Falk Hüffner, Jan 24 2022 */
  • Haskell
    a000695 n = if n == 0 then 0 else 4 * a000695 n' + b
            where (n',b) = divMod n 2
-- Reinhard Zumkeller, Feb 21 2014, Dec 03 2011
  • Julia
    function a(n)
    m, r, b = n, 0, 1
    while m > 0
        m, q = divrem(m, 2)
        r += b * q
        b *= 4
    end
r end; [a(n) for n in 0:51] |> println # Peter Luschny, Jan 03 2021
  • Magma
    m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( (&+[4^k*x^(2^k)/(1+x^(2^k)): k in [0..20]])/(1-x) )); // G. C. Greubel, Dec 06 2018
  • Maple
    a:= proc(n) local m, r, b; m, r, b:= n, 0, 1;
      while m>0 do r:= r+b*irem(m, 2, 'm'); b:= b*4 od; r
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 16 2013
  • Mathematica
    Table[FromDigits[Riffle[IntegerDigits[n, 2], 0], 2], {n, 0, 51}] (* Jacob A. Siehler, Jun 30 2010 *)
Table[FromDigits[IntegerDigits[n, 2], 4], {n, 0, 51}] (* IWABUCHI Yu(u)ki, Apr 06 2013 *)
Union@ Flatten@ NestList[ Join[ 4#, 4# + 1] &, {0}, 6] (* Robert G. Wilson v, Aug 30 2014 *)
Select[ Range[0, 1320], Total@ IntegerDigits[#, 2] == Total@ IntegerDigits[#, 4] &] (* Robert G. Wilson v, Oct 24 2014 *)
Union[FromDigits[#,4]&/@Flatten[Table[Tuples[{0,1},n],{n,6}],1]] (* Harvey P. Dale, Oct 03 2015 *)
a[ n_] := Which[n < 1, 0, EvenQ[n], a[n/2] 4, True, a[n - 1] + 1]; (* Michael Somos, Nov 30 2016 *)
  • PARI
    a(n)=n=binary(n);sum(i=1,#n,n[i]*4^(#n-i)) \\ Charles R Greathouse IV, Mar 04 2013
  • PARI
    {a(n) = if( n<1, 0, n%2, a(n-1) + 1, a(n/2) * 4)}; /* Michael Somos, Nov 30 2016 */
  • PARI
    A000695(n)=fromdigits(binary(n),4) \\ M. F. Hasler, Oct 16 2018
  • Python
    def a(n):
    n = bin(n)[2:]
    x = len(n)
    return sum(int(n[i]) * 4**(x - 1 - i) for i in range(x))
[a(n) for n in range(101)] # Indranil Ghosh, Jun 25 2017
  • Python
    def a():
    x = 0
    while True:
        yield x
        y = ~(x << 1)
        x = (x - y) & y # Falk Hüffner, Dec 21 2021
  • Python
    from itertools import count, islice
def A000695_gen(): # generator of terms
    yield (a:=0)
    for n in count(1):
        yield (a := a+((1<<((~n & n-1).bit_length()<<1)+1)+1)//3)
A000695_list = list(islice(A000695_gen(),30)) # Chai Wah Wu, Feb 22 2023
  • Python
    def A000695(n): return int(bin(n)[2:],4) # Chai Wah Wu, Aug 21 2023
  • Sage
    s=(sum(4^k*x^(2^k)/(1+x^(2^k)) for k in range(10))/(1-x)).series(x, 60); s.coefficients(x, sparse=False) # G. C. Greubel, Dec 06 2018

Formula

G.f.: 1/(1-x) * Sum_{k>=0} 4^k*x^2^k/(1+x^2^k). - Ralf Stephan, Apr 27 2003
Numbers k such that the coefficient of x^k is > 0 in Product_{n>=0} 1+x^(4^n). - Benoit Cloitre, Jul 29 2003
For n >= 1, a(n) = a(n-1) + (4^t+2)/6, where t is such that 2^t||2n,or t=A007814(2n). a(n) = (A145812(n+1) - 1)/2. - Vladimir Shevelev, Nov 07 2008
To get a(n), write n as Sum b_j*2^j, then a(n) = Sum b_j*2^(2j). The Diophantine equation a(k)+2a(l)=n has the unique solution: k=Sum b_(2j)*2^j, l=Sum b_(2j+1)*2^j. - Vladimir Shevelev, Nov 10 2008
If a(k)*a(l)=a(m), then k*l=m (the inverse, generally speaking, is not true). - Vladimir Shevelev, Nov 21 2008
Let F(x) be the generating function, then F(x)*F(x^2) = 1/(1-x). - Joerg Arndt, May 12 2010
a(n+1) = (a(n) + 1/3) & -1/3, where & is bitwise AND, -1/3 is represented as the infinite dyadic ...010101 (just as -1 is ...111111 in two's complement) and +1/3 is ...101011. - Marc LeBrun, Sep 30 2010
a(n) = Sum_{k>=0} {A030308(n,k)*b(k)} with b(k) = 4^k = A000302(k). - Philippe Deléham, Oct 18 2011
A182560(6*a(n)) = 0. - Reinhard Zumkeller, May 05 2012
G.f.: x/(1-x^2) + 4*x^2/((1-x)*(W(0) - 4*x - 4*x^2)), where W(k) = 1 + 4*x^(2^k) + 5*x^(2^(k+1)) - 4*x^(2^(k+1))*(1 + x^(2^(k+1)))^2/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 04 2014
liminf a(n)/n^2 = 1/3 and limsup a(n)/n^2 = 1. - Gheorghe Coserea, Sep 15 2015
Let f(x) = (Sum_{k=-oo..oo} floor(x*2^k)/4^k)/2. Then f(x) is a real-valued extension of a(n), which a(n) approximates in the sense that f(x) = lim_{k->oo} a(floor(x*2^k))/a(2^k). - Velin Yanev, Nov 28 2016
G.f. A(x) satisfies x/(1 - x^2) = A(x) - 4 * (1+x) * A(x^2). - Michael Somos, Nov 30 2016
a(2^k) = 4^k = A000302(k). a(n + 2^k) = a(n) + a(2^k) for 2^k > n >= 1. - David A. Corneth, Oct 16 2018
Sum_{n>=1} 1/a(n) = 1.886176434476107244547259512076353532930680508099044818673061351780360211128... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 12 2022

A075158 Prime factorization of n+1 encoded with the run lengths of binary expansion.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 10, 7, 6, 11, 21, 8, 42, 20, 9, 15, 85, 12, 170, 23, 22, 43, 341, 16, 13, 84, 14, 40, 682, 19, 1365, 31, 41, 171, 18, 24, 2730, 340, 86, 47, 5461, 44, 10922, 87, 17, 683, 21845, 32, 26, 27, 169, 168, 43690, 28, 45, 80, 342, 1364, 87381, 39, 174762
Offset: 0

Views

Author

Antti Karttunen, Sep 13 2002

Keywords

Comments

a(2n) = 1 or 2 mod 4 and a(2n+1) = 0 or 3 mod 4 for all n > 1

Examples

			a(1) = 1 as 2 = 2^1, a(2) = 2 (10 in binary) as 3 = 3^1 * 2^0, a(3) = 3 (11) as 4 = 2^2, a(4) = 5 (101) as 5 = 5^1 * 3^0 * 2^0, a(5) = 4 (100) as 6 = 3^1 * 2^1, a(8) = 6 (110) as 9 = 3^2 * 2^0, a(11) = 8 (1000) as 12 = 3^1 * 2^2, a(89) = 35 (100011) as 90 = 5^1 * 3^2 * 2^1, a(90) = 90 (1011010) as 91 = 13^1 * 11^0 * 7^1 * 5^0 * 3^0 * 2^0.
The binary expansion of a(n) begins from the left with as many 1's as is the exponent of the largest prime present in the factorization of n+1 and from then on follows runs of ej+1 zeros and ones alternatively, where ej are the corresponding exponents of the successively lesser primes (0 if that prime does not divide n+1).

Links

Crossrefs

Inverse of A075157. a(n) = A075160(n+1)-1. a(A006093(n)) = A000975(n). Cf. A059884.

Programs

  • Haskell
    import Data.List (elemIndex); import Data.Maybe (fromJust)
a075158 = fromJust . (`elemIndex` a075157_list)
-- Reinhard Zumkeller, Aug 04 2014

A075173 Prime factorization of n encoded by interleaving successive prime exponents in unary to bit-positions given by columns of A075300.

Original entry on oeis.org

0, 1, 2, 5, 8, 3, 128, 21, 34, 9, 32768, 7, 2147483648, 129, 10, 85, 9223372036854775808, 35, 170141183460469231731687303715884105728, 13, 130, 32769
Offset: 1

Views

Author

Antti Karttunen, Sep 13 2002

Keywords

Comments

As in A059884, here also we store the exponent e_i of p_i (p1=2, p2=3, p3=5, ...) in the factorization of n to the bit positions given by the column i-1 of A075300 (the exponent of 2 is thus stored to bit positions 0, 2, 4, ..., exponent of 3 to 1, 5, 9, 13, ..., exponent of 5 to 3, 11, 19, 27, 35, ...), but using unary instead of binary system, i.e. we actually store 2^(e_i) - 1 in binary.
This injective mapping from N to N offers an example of the proof given in Cameron's book that any distributive lattice can be represented as a sublattice of the power-set lattice P(X) of some set X. This allows us to implement GCD (A003989) with bitwise AND (A004198) and LCM (A003990) with bitwise OR (A003986). Also, to test whether x divides y, it is enough to check that ((a(x) OR a(y)) XOR a(y)) = A003987(A003986(a(x),a(y)),a(y)) is zero.

Examples

			a(24) = 23 because 24 = 2^3 * 3^1 so we add the binary words 10101 and 10 to get 10111 in binary = 23 in decimal and a(25) = 2056 because 25 = 5^2 so we form a binary word 100000001000 = 2056 in decimal.

References

  • P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1998, page 191. (12.3. Distributive lattices)

Crossrefs

Variant: A075175. Inverse: A075174. Cf. A059884.
A003989(x, y) = A075174(A004198(a(x), a(y))), A003990(x, y) = A075174(A003986(a(x), a(y))).

A059900 Successively interleaved alternate bits in binary expansion of n gives vector of exponents in prime factorization of a(n).

Original entry on oeis.org

1, 2, 3, 6, 4, 8, 12, 24, 5, 10, 15, 30, 20, 40, 60, 120, 16, 32, 48, 96, 64, 128, 192, 384, 80, 160, 240, 480, 320, 640, 960, 1920, 9, 18, 27, 54, 36, 72, 108, 216, 45, 90, 135, 270, 180, 360, 540, 1080, 144, 288, 432, 864, 576, 1152, 1728, 3456, 720, 1440, 2160
Offset: 0

Views

Author

Marc LeBrun, Feb 07 2001

Keywords

Comments

Every value appears once. Inverse of A059884. Similar to A052330 (they diverge at n=16) but assigns bits via interleaving versus numerical order.

Examples

			To find a(7): 7 = ...000111; ... 0 0 1 1 = 3 -> 2^3 = 8; ... 0 1 = 1 -> 3^1 = 3; so a(7) = 3*8 = 24.

Links

Crossrefs

Cf. A059884, A052330.

A075302 Transpose of array A075300.

Original entry on oeis.org

0, 2, 1, 4, 5, 3, 6, 9, 11, 7, 8, 13, 19, 23, 15, 10, 17, 27, 39, 47, 31, 12, 21, 35, 55, 79, 95, 63, 14, 25, 43, 71, 111, 159, 191, 127, 16, 29, 51, 87, 143, 223, 319, 383, 255, 18, 33, 59, 103, 175, 287, 447, 639, 767, 511, 20, 37, 67, 119, 207, 351, 575, 895, 1279
Offset: 0

Views

Author

Antti Karttunen, Sep 12 2002

Keywords

Comments

The top-left corner of the array looks like:
0 2 4 6 8 10 12 14 ...
1 5 9 13 17 21 25 29 ...
3 11 19 27 35 43 51 59 ...
7 23 39 55 71 87 ...
15 47 79 ...
This can be used to construct mappings like A059884 and A075173, where the elements of the first row give the bit-positions where to interleave the first element of a finite vector of nonnegative integers, the second row the bit-positions for the second element, etc.

Links

Crossrefs

Inverse permutation: A075303. Transpose: A075300. The X-projection is given by A025480 and the Y-projection is A007814(n+1). Cf. A002262, A025581.

Programs

A059904 Periodic part of continued fraction for sqrt(n), encoded by recursively interleaving the bits in the binary expansions of the repeating terms.

Original entry on oeis.org

0, 4, 33, 0, 16, 516, 549755813899, 513, 0, 20, 549, 548, 604462909948052075708555, 549764202537, 545, 0, 64, 8208, 13479973333575319897333507543509815336818572211270286381289293482126
Offset: 1

Views

Author

Marc LeBrun, Feb 07 2001

Keywords

Comments

Could be made less gigantic by omitting final terms in continued fraction, which are always 2*c0.

Examples

			sqrt(3)=1+[1,2] so a(3) is encoded as:
.....0 0 0 0 1 -> 1
....... 1 . 0 .-> 2
--------------
.....000100001 = 33.

Crossrefs

A059903, A059884, A003285.

Formula

a(n) = A059884(A059903(n)).

A059902 Partitions encoded by interleaving bits in parts. The partition [P1+P2+P3+...] with P1>=P2>=P3>=... is encoded in binary by recursively interleaving the bits of P1 with the (recursively interleaved bits of P2 with the (recursively...)).

Original entry on oeis.org

0, 1, 4, 3, 5, 6, 11, 16, 7, 36, 14, 139, 17, 18, 37, 15, 44, 142, 32907, 20, 19, 48, 39, 26, 45, 2084, 143, 172, 32910, 2147516555, 21, 22, 49, 50, 27, 56, 47, 2085, 154, 173, 2212, 32911, 32940, 2147516558, 9223372039002292363, 64, 23, 52, 51, 528, 30, 57
Offset: 0

Views

Author

Marc LeBrun, Feb 07 2001

Keywords

Comments

Partitions are ordered canonically (as described in the OEIS Wiki link): [] [1] [2] [1+1] [3] [2+1] [1+1+1] [4]...

Examples

			Partition for n=17 is [2+2+1], so a(17) is given by
.....0 0 0 0 0 1 0- -> 2
..... . 0 . 1 . 0 - -> 2
..... 0 ..... 1 ... -> 1
------------------
.....0000000101100 = 44.

Links

Crossrefs

Cf. A059901, A059884, A000041.

Formula

a(n) = A059884(A059901(n)).

Extensions

Terms reordered by Sean A. Irvine, Oct 17 2022
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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