A028897 -id:A028897 - cp's OEIS frontend

A322000 Nonnegative integers, sorted by increasing value of A028897(n) = Sum d[i]2^i for n = Sum d[i]10^i, then value of n.

0, 1, 2, 10, 3, 11, 4, 12, 20, 100, 5, 13, 21, 101, 6, 14, 22, 30, 102, 110, 7, 15, 23, 31, 103, 111, 8, 16, 24, 32, 40, 104, 112, 120, 200, 1000, 9, 17, 25, 33, 41, 105, 113, 121, 201, 1001, 18, 26, 34, 42, 50, 106, 114, 122, 130, 202, 210, 1002, 1010, 19, 27

Offset: 0

M. F. Hasler, Feb 13 2019

A028897(n) is the result of using the decimal digits of n, but weighting their position as in base 2. For sake of brevity we refer to this as the b-value of n in the sequel. This idea is found on the website given in links under the name "decibinary numbers".

The b-values increment by 1 at indices (of "records") 1, 2, 4, 6, 10, 14, 20, 26, 36, ... Prefixing an initial 0, the gaps between these, equal to the number of occurrences of a given b-value (0, 1, 2, ...), are 1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 13, 13, ... = A072170(n,10). In this sequence each of (1, 2, 4, 6, 10, 13, 18, ...) is repeated twice.

			The first terms of the sequence are as follows: (b = A028897)
  n | 0 | 1 | 2 | 10 | 3 | 11 | 4 | 12 | 20 | 100 | 5 | 13 | 21 | 101 | ...
----+---+---+---+----+---+----+---+----+----+-----+---+----+----+-----+-----
b(n)| 0 | 1 | 2 |  2 | 3 |  3 | 4 |  4 |  4 |  4  | 5 |  5 |  5 |  5  | ...
For example, b(345) = 3*2^2 + 4*2 + 5 = 25.

Robert Israel, Table of n, a(n) for n = 0..10000
Robin C. Yu, Decibinary Numbers, on Hackerrank.com.

Cf. A028897, A072170 (see comments).

N:= 30: # for all numbers with A028897(n) <= N
L:= {seq([i,i],i=0..9)}: Agenda:= {seq([i,i],i=1..9)}:
extend:= proc(p) local x;  op(select(t -> t[2]<=N, [seq([10*p[1]+x, 2*p[2]+x],x=0..9)])); end proc:
sorter:= proc(p1,p2) if p1[2] <> p2[2] then p1[2] < p2[2] else p1[1] < p2[1] fi end proc:
while Agenda <> {} do
  Agenda:= map(extend, Agenda);
  L:= L union Agenda;
od:
L:= sort( convert(L,list),sorter):
map(t -> t[1], L); # Robert Israel, Feb 24 2019

my(A028897(n)=fromdigits(digits(n),2),S=[]);for(k=1,2^10,(t=A028897(k))>9||S=setunion(S,[[t,k]]));apply(t->t[2],S)

A322003 Indices where A028897(A322000(n)) increases. Partial sums of A072170(n,10).

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 59, 72, 90, 108, 130, 152, 182, 212, 248, 284, 329, 374, 426, 478, 542, 606, 678, 750, 834, 918, 1011, 1104, 1214, 1324, 1446, 1568, 1708, 1848, 2002, 2156, 2333, 2510, 2702, 2894, 3108, 3322, 3552, 3782, 4040, 4298, 4575, 4852, 5156, 5460, 5784, 6108, 6464, 6820, 7196, 7572, 7977, 8382

Offset: 0

M. F. Hasler, Feb 19 2019

nonn

A322000 lists all nonnegative integers m ordered by increasing "decibinary" value N = A028897(m) = Sum d[i]*2^i where d[i] are the decimal digits of m. A072170(N,10) says in how many ways a given N can be written in that way. Accordingly, this is also the length of runs of identical values A028897(A322000(k)), and the partial sums, listed here as a(k), give the indices of A322000 where the decibinary value of the terms go up by one.

We have a(k) <= A000123(k-1) with equality for 1 <= k <= 10: the first differences of A000123 give back that sequence with terms duplicated, and this is the limiting column of A072170.

Cf. A322000, A028897, A072170.

A322003(n)=sum(k=0,n-1,A072170(k,10))
A322003_vec=vector(99,k,s=if(k>1,s)+A072170(k-1,10)) \\ more efficient for computing a large vector. Excludes the initial a(0) = 0 to have 1-based indices of the vector match the indices of the components a(n), n >= 1.

a(n) = Sum_{0 <= k < n} A072170(k,10).

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100, 10101, 10110, 10111, 11000, 11001, 11010, 11011, 11100, 11101, 11110, 11111, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 100111

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

List of binary numbers. (This comment is to assist people searching for that particular phrase. - N. J. A. Sloane, Apr 08 2016)
Or, numbers that are sums of distinct powers of 10.
Or, numbers having only digits 0 and 1 in their decimal representation.
Complement of A136399; A064770(a(n)) = a(n). - Reinhard Zumkeller, Dec 30 2007
From Rick L. Shepherd, Jun 25 2009: (Start)
Nonnegative integers with no decimal digit > 1.
Thus nonnegative integers n in base 10 such that kn can be calculated by normal addition (i.e., n + n + ... + n, with k n's (but not necessarily k + k + ... + k, with n k's)) or multiplication without requiring any carry operations for 0 <= k <= 9. (End)
For n > 1: A257773(a(n)) = 10, numbers that are Belgian-k for k=0..9. - Reinhard Zumkeller, May 08 2015
For any integer n>=0, find the binary representation and then interpret as decimal representation giving a(n). - Michael Somos, Nov 15 2015
N is in this sequence iff A007953(N) = A101337(N). A028897 is a left inverse. - M. F. Hasler, Nov 18 2019
For n > 0, numbers whose largest decimal digit is 1. - Stefano Spezia, Nov 15 2023

			a(6)=110 because (1/2)*((1-(-1)^6)*10^0 + (1-(-1)^3)*10^1 + (1-(-1)^1)*10^2) = 10 + 100.
G.f. = x + 10*x^2 + 11*x^3 + 100*x^4 + 101*x^5 + 110*x^6 + 111*x^7 + 1000*x^8 + ...
.
  000    The numbers < 2^n can be regarded as vectors with
  001    a fixed length n if padded with zeros on the left
  010    side. This represents the n-fold Cartesian product
  011    over the set {0, 1}. In the example on the left,
  100    n = 3. (See also the second Python program.)
  101    Binary vectors in this format can also be seen as a
  110    representation of the subsets of a set with n elements.
  111    - _Peter Luschny_, Jan 22 2024

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 21.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §2.8 Binary, Octal, Hexadecimal, p. 64.
Manfred R. Schroeder, "Fractals, Chaos, Power Laws", W. H. Freeman, 1991, p. 383.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 0..32768 (first 8192 terms from Franklin T. Adams-Watters)
Heinz Gumin, Herrn von Leibniz' Rechnung mit Null und Eins, Siemens AG, 3. Auflage 1979 -- contains facsimiles of Leibniz's papers from 1679 and 1703.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.
G. W. Leibniz, Explication de l'arithmétique binaire, qui se sert des seuls caractères 0 & 1; avec des remarques sur son utilité, et sur ce qu'elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de l'Académie Royale des Sciences, 1703, pp. 85-89; reprinted in Gumin (1979).
N. J. A. Sloane, Table of a(n) for n = 0..1048576 (A large file).
Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.
Index entries for sequences related to Most Wanted Primes video.
Index entries for 10-automatic sequences.
Index entries for sequences related to binary expansion of n.

The basic sequences concerning the binary expansion of n are this one, A000120 (Hammingweight: sum of bits), A000788 (partial sums of A000120), A000069 (A000120 is odd), A001969 (A000120 is even), A023416 (number of bits 0), A059015 (partial sums). Bisections A099820 and A099821.
Cf. A028897 (convert binary to decimal).
Cf. A000042, A007089-A007095, A000695, A005836, A033042-A033052, A159918, A004290, A169965, A169966, A169967, A169964, A204093, A204094, A204095, A097256, A257773, A257770.
Cf. A000290, A001737, A007953, A030308, A054055, A064770, A101337, A136399.

a007088 0 = 0
a007088 n = 10 * a007088 n' + m where (n',m) = divMod n 2
-- Reinhard Zumkeller, Jan 10 2012

A007088 := n-> convert(n, binary): seq(A007088(n), n=0..50); # R. J. Mathar, Aug 11 2009

Table[ FromDigits[ IntegerDigits[n, 2]], {n, 0, 39}]
Table[Sum[ (Floor[( Mod[f/2 ^n, 2])])*(10^n) , {n, 0, Floor[Log[2, f]]}], {f, 1, 100}] (* José de Jesús Camacho Medina, Jul 24 2014 *)
FromDigits/@Tuples[{1,0},6]//Sort (* Harvey P. Dale, Aug 10 2017 *)

{a(n) = subst( Pol( binary(n)), x, 10)}; /* Michael Somos, Jun 07 2002 */

{a(n) = if( n<=0, 0, n%2 + 10*a(n\2))}; /* Michael Somos, Jun 07 2002 */

a(n)=fromdigits(binary(n),10) \\ Charles R Greathouse IV, Apr 08 2015

def a(n): return int(bin(n)[2:])
print([a(n) for n in range(40)]) # Michael S. Branicky, Jan 10 2021

from itertools import product
n = 4
for p in product([0, 1], repeat=n): print(''.join(str(x) for x in p))
# Peter Luschny, Jan 22 2024

a(n) = Sum_{i=0..m} d(i)*10^i, where Sum_{i=0..m} d(i)*2^i is the base 2 representation of n.
a(n) = (1/2)*Sum_{i>=0} (1-(-1)^floor(n/2^i))*10^i. - Benoit Cloitre, Nov 20 2001
a(n) = A097256(n)/9.
a(2n) = 10*a(n), a(2n+1) = a(2n)+1.
G.f.: 1/(1-x) * Sum_{k>=0} 10^k * x^(2^k)/(1+x^(2^k)) - for sequence as decimal integers. - Franklin T. Adams-Watters, Jun 16 2006
a(A000290(n)) = A001737(n). - Reinhard Zumkeller, Apr 25 2009
a(n) = Sum_{k>=0} A030308(n,k)*10^k. - Philippe Deléham, Oct 19 2011
For n > 0: A054055(a(n)) = 1. - Reinhard Zumkeller, Apr 25 2012
a(n) = Sum_{k=0..floor(log_2(n))} floor((Mod(n/2^k, 2)))*(10^k). - José de Jesús Camacho Medina, Jul 24 2014

A010062 a(0)=1; thereafter a(n+1) = a(n) + number of 1's in binary representation of a(n).

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 12, 14, 17, 19, 22, 25, 28, 31, 36, 38, 41, 44, 47, 52, 55, 60, 64, 65, 67, 70, 73, 76, 79, 84, 87, 92, 96, 98, 101, 105, 109, 114, 118, 123, 129, 131, 134, 137, 140, 143, 148, 151, 156, 160, 162, 165, 169, 173, 178, 182, 187, 193, 196, 199, 204

Offset: 0

Leonid Broukhis, Mar 15 1996

Sequence A230297 (and A157845 without initial term) converted from binary to decimal, cf. formula. - M. F. Hasler, Nov 18 2019

			a(7) = 14 because a(6) = 12, which is 1100 in binary (having 2 on bits), and 12 + 2 = 14.
a(8) = 17 because a(7) = 14, which is 1110 in binary (having 3 on bits), and 14 + 3 = 17.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Raoul Nakhmanson-Kulish, Graph of a(n)/(n*log_2(n)/2), showing self-similar fractal structure.
Raoul Nakhmanson-Kulish, Graph of f(n), where f(n) = (a(n)-n*log_2(n)/2)/(n*sqrt(log_2(n)*log_2 log_2(n))) (see Stolarsky's estimate below).
Kenneth B. Stolarsky, The sum of a digitaddition series, Proc. Amer. Math. Soc. 59 (1976), no. 1, 1--5. MR0409340 (53 #13099)
Index entries for Colombian or self numbers and related sequences

First row of A228083.
For the base-10 analog see A004207.
Cf. A000120, A010061, A092391, A229167, A096303, A229743, A229744, A230297 (this sequence written in binary), A230298 (read mod 2).
See A230088 for partial sums.
Equals A028897 o A230297 = A028897 o A157845 (up to offset); see also A007088.

a010062 n = a010062_list !! n
a010062_list = iterate a092391 1  -- Reinhard Zumkeller, May 13 2012

[n le 1 select 1 else Self(n-1)+&+Intseq(Self(n-1),2): n in [1..61]]; // Bruno Berselli, Oct 27 2012

NestList[# + DigitCount[#, 2, 1] &, 1, 60] (* Alonso del Arte, Oct 26 2012 *)

print1(s=1);for(n=2,30,print1(", ", s+=hammingweight(s))) \\ Charles R Greathouse IV, Oct 27 2012

A010062=List(1); A010062(n)={for(n=#A010062,n, listput(A010062, A092391(A010062[n])));A010062[n+1]} \\ A092391(n)=n+hammingweight(n) \\ M. F. Hasler, Nov 18 2019

from itertools import islice
def agen():
    an = 1
    while True: yield an; an += an.bit_count()
print(list(islice(agen(), 61))) # Michael S. Branicky, Jul 31 2022

a(n) = (n/2)*log n + O(n*sqrt(log n * loglog n)), where log means log_2. In particular, a(n) ~ (n/2)*log n. [Stolarsky]
a(n + 1) = A092391(a(n)) = a(n) + A000120(a(n)). - Reinhard Zumkeller, May 27 2012, May 08 2004; corrected thanks to a notice by Lambert Herrgesell
a(n) = A028897(A230297(n)) = A028897(A157845(n+1)). - M. F. Hasler, Nov 18 2019

More terms from Benoit Cloitre, Jun 02 2002

Stolarsky reference from Matthew C. Russell, Oct 08 2013

A122618 a(n) = n_n, where "N_b" denotes "N read in base b": if N = Sum c_i 10^i then N_b = Sum c_i b^i.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 250, 256, 262, 268, 274, 280, 286, 292, 298, 304, 360, 367, 374, 381, 388

Offset: 1

N. J. A. Sloane, Sep 21 2006

The definition applies even if b < 10. Examples: 23_45 = 2*45 + 3 = 93, 23_2 = 2*2 + 3 = 7.

David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 393-402.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Descending Dungeons and Iterated Base-Changing, arXiv:math/0611293 [math.NT], 2006-2007.
David Applegate, Marc LeBrun, N. J. A. Sloane, Descending Dungeons, Problem 11286, Amer. Math. Monthly, 116 (2009) 466-467.

Differs from A083292 starting at n=100.

Cf. A028897 (n_2), A122640 (2n_2).

A122618 := proc(n)
      local dgs;
      dgs := convert(n,base,10) ;
    add(op(i,dgs)*n^(i-1),i=1..nops(dgs)) ;
end proc: # R. J. Mathar, May 06 2019

f[n_] := FromDigits[ IntegerDigits@n, n]; Array[f, 64] (* Robert G. Wilson v, Sep 27 2006 *)

A122618(n,d=digits(n))=d*vectorv(#d,i,n^(#d-i)) \\ M. F. Hasler, Apr 22 2015

A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 5

Offset: 0

Antti Karttunen, Jun 22 2014

This sequence converts any number from various "Catalan Base number systems" (when represented as decimal numbers) back to the integer the numeral represents: e.g. we have a(A014418(n)) = n and a(A244159(n)) = n (except for the latter this is eventually broken by the shortcomings of the decimal representation used, while for the former it works for all n, because no digits larger than 3 will ever appear in the terms of A014418).
A197433 is similar, but replaces 2^k with A000108(k+1) in binary expansion of n.
For 1- and 2-digit numbers the same as A156230. - R. J. Mathar, Jun 27 2014

Index entries for sequences related to decimal expansion of n

Cf. A000108, A014418, A244159, A197433, A244155, A244156, A244157.

Differs from A028897 and A081594 for the first time at n=100, which here is a(100) = 5.

A244158 := proc(n)
    local dgs,k ;
    dgs := convert(n,base,10) ;
    add( op(k,dgs)*A000108(k),k=1..nops(dgs)) ;
end proc: # R. J. Mathar, Jan 31 2015

A081594 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 2x+y.

Original entry on oeis.org

Offset: 0

N. J. A. Sloane, Apr 22 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,1,-1).

Cf. A081502. Differs from A028897, A156230 and A244158 for the first time at n=100, which here is a(100) = 20.

[(n+4*y)/5 where y is n mod 10: n in [0..100]]; // Bruno Berselli, Jun 24 2014

A081594:=n->n-8*floor(n/10); seq(A081594(n), n=0..100); # Wesley Ivan Hurt, Jun 25 2014

CoefficientList[Series[-x (7 x^9 - x^8 - x^7 - x^6 - x^5 - x^4 - x^3 - x^2 - x - 1)/((x - 1)^2 (x + 1) (x^4 - x^3 + x^2 - x+1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 150}], x] (* Vincenzo Librandi, Jun 25 2014 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,1,-1},{0,1,2,3,4,5,6,7,8,9,2},110] (* or *) Table[Range[n,n+9],{n,0,26,2}]//Flatten (* Harvey P. Dale, Jul 22 2021 *)

my(n, x, y); vector(200, n, y=(n-1)%10; x=(n-1-y)\10; 2*x+y) \\ Colin Barker, Jun 24 2014

[n-8*floor(n/10) for n in (0..100)] # Bruno Berselli, Jun 24 2014

a(n) = (2 * floor(n/10)) + (n modulo 10). - Antti Karttunen, Jun 22 2014
G.f.: -x*(7*x^9 -x^8 -x^7 -x^6 -x^5 -x^4 -x^3 -x^2 -x -1) / ((x -1)^2*(x +1)*(x^4 -x^3 +x^2 -x +1)*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jun 23 2014
a(n) = n - 8*floor(n/10). [Bruno Berselli, Jun 24 2014]

Terms up to n=100 added by Antti Karttunen, Jun 22 2014

G.f. revised by Vincenzo Librandi, Jun 25 2014

A230297 a(n) = A010062(n) written in binary: a(n+1) = a(n) + hammingweight(a(n)) in binary.

Original entry on oeis.org

1, 10, 11, 101, 111, 1010, 1100, 1110, 10001, 10011, 10110, 11001, 11100, 11111, 100100, 100110, 101001, 101100, 101111, 110100, 110111, 111100, 1000000, 1000001, 1000011, 1000110, 1001001, 1001100, 1001111, 1010100, 1010111, 1011100, 1100000, 1100010, 1100101, 1101001, 1101101, 1110010, 1110110, 1111011, 10000001, 10000011

Offset: 0

N. J. A. Sloane, Oct 17 2013

Is there any way to tell by looking at a binary number whether or not it is a term of this sequence?

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Index entries for Colombian or self numbers and related sequences.

Cf. A010062.
Essentially the same as A157845.
Cf. A004207 (base-10 analog); A007088 (n in binary), A010062 (this written in base 10), A000120 (Hammingweight), A092391 (A000120(n) + n), A028897 (convert binary to decimal).

s[0] = 1; s[n_] := s[n] = s[n-1] + DigitCount[s[n-1], 2, 1]; Table[FromDigits[IntegerDigits[s[n], 2]], {n, 0, 50}] (* Amiram Eldar, Jul 28 2023 *)

(A230297(n)=A007088(A010062(n))); A230297_vec(N)={vector(N,i, if(i>1, A007088(N+=hammingweight(N)), N=1))} \\ M. F. Hasler, Nov 18 2019

a(n) = A157845(n+1) = A007088(A010062(n)) = A007088(A092391(A028897(a(n-1)))). - M. F. Hasler, Nov 18 2019

A157845 a(0) = 1, a(n) = sum of binary digits of all prior terms, expressed in binary.

Original entry on oeis.org

1, 1, 10, 11, 101, 111, 1010, 1100, 1110, 10001, 10011, 10110, 11001, 11100, 11111, 100100, 100110, 101001, 101100, 101111, 110100, 110111, 111100, 1000000, 1000001, 1000011, 1000110, 1001001, 1001100, 1001111, 1010100, 1010111, 1011100, 1100000, 1100010

Offset: 0

Oliver K. Seet, Mar 07 2009

Equals A230297 = A010062 converted from decimal to binary, prefixed by another initial 1. - M. F. Hasler, Nov 18 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A004207 (base-10 analog); A007088 (n in binary), A010062 (this written in base 10), A000120 (Hammingweight), A092391 (A000120(n) + n), A028897 (convert binary to decimal).

b:= proc(n) option remember; `if`(n<2, 1, b(n-1)+
      add(i, i=convert(a(n-1), base, 10)))
    end:
a:= n-> convert(b(n), binary):
seq(a(n), n=0..44);  # Alois P. Heinz, Nov 18 2019

s[0] = s[1] = 1; s[n_] := s[n] = s[n-1] + DigitCount[s[n-1], 2, 1]; Table[FromDigits[IntegerDigits[s[n], 2]], {n, 0, 50}] (* Amiram Eldar, Jul 28 2023 *)

lista(nn) = {my(s = 1); my(t = 1); print1(t, ", "); for (i=1, nn, sb = binary(s); t = subst(Pol(sb), x, 10); print1(t, ", "); s += hammingweight(sb););}

apply( A157845(n)=fromdigits(binary(A010062(n-!!n))), [0..40]) \\ M. F. Hasler, Nov 18 2019

a(n) = A230297(n-1) = A007088(A010062(n-1)) = A007088(A092391(A028897(a(n-1)))) for n > 0. - M. F. Hasler, Nov 18 2019

a(11) corrected and extended by R. J. Mathar, Mar 12 2009

More terms from Michel Marcus, Apr 19 2014

A222813 Numbers whose binary representation is palindromic and in which all runs of 0's and 1's have length at least 2.

Original entry on oeis.org

3, 7, 15, 31, 51, 63, 99, 127, 195, 231, 255, 387, 455, 511, 771, 819, 903, 975, 1023, 1539, 1651, 1799, 1935, 2047, 3075, 3171, 3315, 3591, 3687, 3855, 3999, 4095, 6147, 6371, 6643, 7175, 7399, 7695, 7967, 8191, 12291, 12483, 12771, 13107, 13299, 14343, 14535, 14823, 15375, 15567, 15903, 16191, 16383, 24579

Offset: 1

N. J. A. Sloane, Mar 11 2013

These are the decimal representations of A061851 read as base-2 numbers.

The terms with an odd number L = 2k-1 of bits, i.e., 2^(L-1) < a(n) < 2^L, are given by the terms of A033015 with length k, shifted k-1 digits to the left and 'OR'ed with the binary reversal of the term. Terms with an even number L = 2k of digits are given as m*2^k + (binary reversal of m) where m runs over the k-bit terms from A033015 and the k-1 bit terms with the last bit negated appended). This explains the FORMULA for the number of terms of given size. - M. F. Hasler, Oct 17 2022

			51 (base 10) = 110011 (base 2), which is a palindrome and has three runs all of length 2.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 608 terms from N. J. A. Sloane)

Cf. A061851.

Cf. A006995 (binary palindromes), A033015 (no isolated binary digit), A028897 ("rebase" 10 -> 2).

brpalQ[n_]:=Module[{idn2=IntegerDigits[n,2]},idn2==Reverse[idn2] && Min[ Length/@ Split[idn2]]>1]; Select[Range[25000],brpalQ] (* Harvey P. Dale, May 21 2014 *)

is(n)=is_A033015(n)&&Vecrev(n=binary(n))==n \\ M. F. Hasler, Oct 06 2022

{A222813_row(n, s=A033015_row(n\/2))=apply(A030101, if(n%2, s\2, n>2, s=setunion([k*2+1-k%2|k<-A033015_row(n\2-1)],s), s=[1]))+s<<(n\2)} \\ Terms with n bits, i.e. between 2^(n-1) and 2^n. - M. F. Hasler, Oct 17 2022

From M. F. Hasler, Oct 06 2022: (Start)
Intersection of A006995 and A033015: binary palindromes with no isolated digit.
There are A000045(A004526(k)) = Fibonacci(floor(k/2)) terms between 2^(k-1) and 2^k.
a(n) = A028897(A061851(n)), where A028897 = convert binary to decimal. (End)

cp's OEIS Frontend

A322000 Nonnegative integers, sorted by increasing value of A028897(n) = Sum d[i]*2^i for n = Sum d[i]*10^i, then value of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A322003 Indices where A028897(A322000(n)) increases. Partial sums of A072170(n,10).

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

A007088 The binary numbers (or binary words, or binary vectors, or binary expansion of n): numbers written in base 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

PARI

Python

Python

Formula

A010062 a(0)=1; thereafter a(n+1) = a(n) + number of 1's in binary representation of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

PARI

Python

Formula

Extensions

A122618 a(n) = n_n, where "N_b" denotes "N read in base b": if N = Sum c_i 10^i then N_b = Sum c_i b^i.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A244158 If n = Sum c_i * 10^i then a(n) = Sum c_i * Cat(i+1), where Cat(k) = A000108(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

A081594 Let n = 10x + y where 0 <= y <= 9, x >= 0. Then a(n) = 2x+y.

Views

A322000 Nonnegative integers, sorted by increasing value of A028897(n) = Sum d[i]2^i for n = Sum d[i]10^i, then value of n.