A035928 -id:A035928 - cp's OEIS frontend

A278465 Numbers that are the sum of one or two terms of A035928.

2, 4, 10, 12, 14, 20, 22, 24, 38, 40, 42, 44, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 76, 80, 84, 90, 94, 98, 104, 108, 112, 142, 144, 150, 152, 154, 160, 162, 170, 172, 178, 180, 182, 184, 188, 190, 192, 194, 198, 202, 204, 206, 208, 212, 214, 216, 220, 222, 224, 226, 230, 232, 234, 240, 242, 244, 246, 250, 252, 254, 256, 260

Offset: 1

Jeffrey Shallit, Nov 22 2016

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A035928.

N:= 1000: # to get all terms <= N
bcr:= proc(n) local L,m;
  L:= convert(n,base,2);
  m:= nops(L);
  add((1-L[i])*2^(m-i),i=1..m)
end proc:
A035928:= select(`<=`,{seq(seq(x*2^d + bcr(x), x=2^(d-1)..2^d-1),d=1..(ilog2(N)+1)/2)},N):
sort(convert(A035928 union {seq(seq(A035928[i]+t, t = select(`<=`,A035928[i..-1], N-A035928[i])),i=1..nops(A035928))},list)); # Robert Israel, Nov 23 2016

max = 1000;
bcrQ[n_] := Module[{idn2 = IntegerDigits[n, 2]}, Reverse[idn2 /. {1 -> 0, 0 -> 1}] == idn2];
A035928 = Select[Range[max], bcrQ];
Union[Total /@ Tuples[A035928, 2], A035928] // Select[#, # <= max&]& (* Jean-François Alcover, Jul 29 2020, after Harvey P. Dale in A035928 *)

A318569 a(n) is the smallest positive integer such that n*a(n) is a "binary antipalindrome" (i.e., an element of A035928).

Original entry on oeis.org

2, 1, 4, 3, 2, 2, 6, 7, 62, 1, 62, 1, 4, 3, 10, 15, 10, 31, 2, 12, 2, 31, 26, 10, 6, 2, 116, 2, 8, 5, 18, 31, 254, 5, 78, 26, 18, 1, 24, 6, 18, 1, 70, 86, 11894, 13, 254, 5, 46, 3, 4, 1, 4, 58, 264, 1, 850, 4, 162, 4, 16, 9, 34, 63, 38, 127, 56, 3, 42, 39, 2

Offset: 1

Jeffrey Shallit, Oct 12 2018

a(n) exists: write n = r*2^i, where r is odd. Then r divides 2^phi(r) - 1, where phi is the Euler phi function. Choose k such that k phi(r) >= i.

Then n divides (2^{k*phi(r)} - 1)*2^{k*phi(r)}, which is a binary antipalindrome.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A318569

Cf. A035928, A342582.

PARI
```
See Links section.
```

def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
def a(n):
    kn = n
    while BCR(kn) != kn: kn += n
    return kn//n
print([a(n) for n in range(1, 72)]) # Michael S. Branicky, Mar 20 2022

A342582 a(n) is the least multiple of n that is a "binary antipalindrome" (i.e., an element of A035928).

Original entry on oeis.org

2, 2, 12, 12, 10, 12, 42, 56, 558, 10, 682, 12, 52, 42, 150, 240, 170, 558, 38, 240, 42, 682, 598, 240, 150, 52, 3132, 56, 232, 150, 558, 992, 8382, 170, 2730, 936, 666, 38, 936, 240, 738, 42, 3010, 3784, 535230, 598, 11938, 240, 2254, 150, 204, 52, 212, 3132

Offset: 1

Rémy Sigrist, Mar 15 2021

This sequence has similarities with A141709.

			For n = 42:
- 42 is a binary antipalindrome,
- so a(42) = 42.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program for A342582

Cf. A035928, A141709, A318569.

PARI
```
See Links section.
```

def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
def bin_anti_pal(n): return BCR(n) == n
def a(n):
    kn = n
    while not bin_anti_pal(kn): kn += n
    return kn
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Mar 15 2021

a(n) = n * A318569(n).

A351173 Natural numbers that cannot be represented as the quotient of two antipalindromic numbers (A035928).

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 11, 12, 13, 14, 16, 22, 23, 25, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 60, 62, 64, 73, 76, 77, 79, 80, 81, 86, 88, 90, 91, 93, 95, 97, 98, 99, 100, 101, 103

Offset: 1

Jeffrey Shallit, Feb 04 2022

James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.

Cf. A035928. Complement of A351172.

A351175 Natural numbers k such that there are infinitely many representations k = A/B with A, B in A035928.

Original entry on oeis.org

1, 6, 15, 18, 19, 20, 24, 28, 51, 59, 61, 63, 66, 67, 68, 71, 72, 74, 75, 78, 82, 83, 84, 87, 94, 96, 120, 191, 195, 203, 211, 231, 235, 243, 247, 251, 253, 255, 258, 260, 262, 263, 264, 265, 266, 267, 271, 272, 274, 275, 278, 279, 285, 288, 292, 293, 298, 299

Offset: 1

Jeffrey Shallit, Feb 04 2022

This sequence and A351176 form a disjoint partition of A351172.

James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.
James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.

Cf. A035928, A351172, A351176.

A278546 Even numbers that cannot be expressed as a sum of 3 or fewer terms of A035928.

Original entry on oeis.org

8, 18, 28, 130, 134, 138, 148, 158, 176, 318, 530, 538, 548, 576, 644, 1300, 2170, 2202, 2212, 2228, 2230, 2248, 8706, 8938, 8948, 34970, 35082

Offset: 1

Jeffrey Shallit, Nov 23 2016

This is conjectured to be the complete list. There are no other examples smaller than 16773120.

Cf. A035928, A278465.

nn = 10^5; Complement[Range[2, nn, 2], Union@ Map[Total, Rest@ Tuples[{0}~Join~#, 3]] &@ Select[Range@ nn, Function[k, Reverse[# /. {1 -> 0, 0 -> 1}] == # &@ IntegerDigits[k, 2]]]] (* Michael De Vlieger, Nov 23 2016, after Harvey P. Dale at A035928 *)

A351325 Numbers k with exactly one solution to the equation A = B*k, where A and B are antipalindromic numbers (members of A035928).

Original entry on oeis.org

5, 21, 26, 69, 85, 89, 92, 102, 106, 116, 219, 221, 233, 239, 245, 249, 261, 269, 276, 284, 291, 301, 306, 319, 323, 324, 333, 341, 344, 356, 361, 364, 369, 426, 434, 460, 468, 488, 843, 869, 879, 919, 925, 971, 981, 997, 1015, 1042, 1044, 1046, 1052, 1053

Offset: 1

Jeffrey Shallit, Feb 07 2022

			For k = 233 we have A = 8532926, B = 36622, which is the only solution in antipalindromes.

James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, arXiv:2202.13694 [math.NT], 2022.

Cf. A002450, A035928, A351172, A351176.

A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

Original entry on oeis.org

1, 1, 3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, 16365, 32640, 65535, 130788, 262143, 523770, 1048509, 2096127, 4194303, 8386440, 16777200, 33550335, 67108608, 134209530, 268435455, 536854005, 1073741823, 2147450880

Offset: 1

N. J. A. Sloane

Also number of compositions of n into relatively prime parts (that is, the gcd of all the parts is 1). Also number of subsets of {1,2,..,n} containing n and consisting of relatively prime numbers. - Vladeta Jovovic, Aug 13 2003
Also number of perfect parity patterns that have exactly n columns (see A118141). - Don Knuth, May 11 2006
a(n) is odd if and only if n is squarefree (Tim Keller). - Emeric Deutsch, Apr 27 2007
a(n) is a multiple of 3 for all n>=3 (see Problem 11161 link). - Emeric Deutsch, Aug 13 2008
Row sums of triangle A143424. - Gary W. Adamson, Aug 14 2008
a(n) is the number of monic irreducible polynomials with nonzero constant coefficient in GF(2)[x] of degree n. - Michel Marcus, Oct 30 2016
a(n) is the number of aperiodic compositions of n, the number of compositions of n with relatively prime parts, and the number of compositions of n with relatively prime run-lengths. - Gus Wiseman, Dec 21 2017

			For n=4, there are 6 compositions of n into coprime parts: <3,1>, <2,1,1>, <1,3>, <1,2,1>, <1,1,2>, and <1,1,1,1>.
From _Gus Wiseman_, Dec 19 2017: (Start)
The a(6) = 27 aperiodic compositions are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1221), (1311), (2112), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
The a(6) = 27 compositions into relatively prime parts are:
  (111111),
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1122), (1131), (1212), (1221), (1311), (2112), (2121), (2211), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (51).
The a(6) = 27 compositions with relatively prime run-lengths are:
  (11112), (11121), (11211), (12111), (21111),
  (1113), (1131), (1212), (1221), (1311), (2112), (2121), (3111),
  (114), (123), (132), (141), (213), (231), (312), (321), (411),
  (15), (24), (42), (51),
  (6).
(End)

H. O. Peitgen and P. H. Richter, The Beauty of Fractals, Springer-Verlag; contribution by A. Douady, p. 165.
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).

Seiichi Manyama, Table of n, a(n) for n = 1..3322 (terms 1..300 from T. D. Noe)
Hunki Baek, Sejeong Bang, Dongseok Kim, and Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014. See Table 2.
Donald Knuth, Robin Chapman and Reiner Martin, Problem 11243, Perfect Parity Patterns, Am. Math. Monthly 115 (7) (2008) p 668, function c(n).
Emeric Deutsch and Lafayette College Problem Group, Problem 11161: Compositions without Common Factors, American Mathematical Monthly, vol. 114, No. 4, 2007, p. 363.
H. W. Gould, Binomial coefficients, the bracket function and compositions with relatively prime summands, Fib. Quart. 2(4) (1964), 241-260.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 2.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
Nicolae Mihalache and Francois Vigneron, Factorization of the quadratic Misiurewicz-Thurston polynomials, arXiv:2506.17662 [math.DS], 2025. See p. 8.
Robert Munafo, Enumeration of Period-N Mu-Atoms
Jeffrey Shallit and N. J. A. Sloane, Correspondence 1974-1975
François Vigneron and Nicolae Mihalache, How to split a tera-polynomial, arXiv:2402.06083 [math.NA], 2024.
Index entries for sequences related to Lyndon words

Cf. A000837, A003239, A008683, A008965, A022553, A034738, A035928, A038199, A051168, A054525, A056267, A059966, A143424, A167606, A178472, A216954, A228369, A294859, A296302.
Equals A027375/2.
See A056278 for a variant.
First differences of A085945.
Column k=2 of A143325.
Row sums of A101391.

with(numtheory): a[1]:=1: a[2]:=1: for n from 3 to 32 do div:=divisors(n): a[n]:=2^(n-1)-sum(a[n/div[j]],j=2..tau(n)) od: seq(a[n],n=1..32); # Emeric Deutsch, Apr 27 2007
with(numtheory); A000740:=n-> add(mobius(n/d)*2^(d-1), d in divisors(n)); # N. J. A. Sloane, Oct 18 2012

a[n_] := Sum[ MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Feb 03 2012, after PARI *)

a(n) = sumdiv(n,d,moebius(n/d)*2^(d-1))

from sympy import mobius, divisors
def a(n): return sum([mobius(n // d) * 2**(d - 1) for d in divisors(n)])
[a(n) for n in range(1, 101)]  # Indranil Ghosh, Jun 28 2017

a(n) = Sum_{d|n} mu(n/d)*2^(d-1), Mobius transform of A011782. Furthermore, Sum_{d|n} a(d) = 2^(n-1).
a(n) = A027375(n)/2 = A038199(n)/2.
a(n) = Sum_{k=0..n} A051168(n,k)*k. - Max Alekseyev, Apr 09 2013
Recurrence relation: a(n) = 2^(n-1) - Sum_{d|n,d>1} a(n/d). (Lafayette College Problem Group; see the Maple program and Iglesias eq (6)). - Emeric Deutsch, Apr 27 2007
G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 2*x^k). - Ilya Gutkovskiy, Oct 24 2018
G.f. satisfies Sum_{n>=1} A( (x/(1 + 2*x))^n ) = x. - Paul D. Hanna, Apr 02 2025

Connection with Mandelbrot set discovered by Warren D. Smith and proved by Robert Munafo, Feb 06 2000

Ambiguous term a(0) removed by Max Alekseyev, Jan 02 2012

A036044 BCR(n): write in binary, complement, reverse.

Original entry on oeis.org

1, 0, 2, 0, 6, 2, 4, 0, 14, 6, 10, 2, 12, 4, 8, 0, 30, 14, 22, 6, 26, 10, 18, 2, 28, 12, 20, 4, 24, 8, 16, 0, 62, 30, 46, 14, 54, 22, 38, 6, 58, 26, 42, 10, 50, 18, 34, 2, 60, 28, 44, 12, 52, 20, 36, 4, 56, 24, 40, 8, 48, 16, 32, 0, 126, 62, 94, 30, 110, 46, 78, 14, 118, 54, 86

Offset: 0

N. J. A. Sloane

a(0) could be considered to be 0 if the binary representation of zero were chosen to be the empty string. - Jason Kimberley, Sep 19 2011
From Bernard Schott, Jun 15 2021: (Start)
Except for a(0) = 1, every term is even.
For each q >= 0, there is one and only one odd number h such that a(n) = 2*q iff n = h*2^m-1 for m >= 1 when q = 0, and for m >= 0 when q >= 1 (see A345401 and some examples below).
a(n) =  0 iff n =    2^m-1 for m >= 1 (Mersenne numbers) (A000225).
a(n) =  2 iff n =  3*2^m-1 for m >= 0 (A153893).
a(n) =  4 iff n =  7*2^m-1 for m >= 0 (A086224).
a(n) =  6 iff n =  5*2^m-1 for m >= 0 (A153894).
a(n) =  8 iff n = 15*2^m-1 for m >= 0 (A196305).
a(n) = 10 iff n = 11*2^m-1 for m >= 0 (A086225).
a(n) = 12 iff n = 13*2^m-1 for m >= 0 (A198274).
For k >= 1, a(n) = 2^k iff n = (2^(k+1)-1)*2^m - 1 for m >= 0.
Explanation for a(n) = 2:
   For m >= 0, A153893(m) = 3*2^m-1 -> 1011...11 -> 0100...00 -> 10 -> 2 where 1011...11_2 is 10 followed by m 1's. (End)

			4 -> 100 -> 011 -> 110 -> 6.

Indranil Ghosh, Table of n, a(n) for n = 0..10000 (first 1024 terms from T. D. Noe)

Cf. A030101, A056539, A329369, A345401.
Cf. A035928 (fixed points), A195063, A195064, A195065, A195066.
Indices of terms 0, 2, 4, 6, 8, 10, 12, 14, 18, 22, 26, 30: A000225 \ {0}, A153893, A086224, A153894, A196305, A086225, A198274, A052996\{1,3}, A291557, A198276, A171389, A198275.

import Data.List (unfoldr)
a036044 0 = 1
a036044 n = foldl (\v d -> 2 * v + d) 0 (unfoldr bc n) where
   bc 0 = Nothing
   bc x = Just (1 - m, x') where (x',m) = divMod x 2
-- Reinhard Zumkeller, Sep 16 2011

A036044:=func; // Jason Kimberley, Sep 19 2011

A036044 := proc(n)
    local bcr ;
    if n = 0 then
        return 1;
    end if;
    convert(n,base,2) ;
    bcr := [seq(1-i,i=%)] ;
    add(op(-k,bcr)*2^(k-1),k=1..nops(bcr)) ;
end proc:
seq(A036044(n),n=0..200) ; # R. J. Mathar, Nov 06 2017

dtn[ L_ ] := Fold[ 2#1+#2&, 0, L ]; f[ n_ ] := dtn[ Reverse[ 1-IntegerDigits[ n, 2 ] ] ]; Table[ f[ n ], {n, 0, 100} ]
Table[FromDigits[Reverse[IntegerDigits[n,2]/.{1->0,0->1}],2],{n,0,80}] (* Harvey P. Dale, Mar 08 2015 *)

a(n)=fromdigits(Vecrev(apply(n->1-n,binary(n))),2) \\ Charles R Greathouse IV, Apr 22 2015

def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def BCR(n): return int(comp(bin(n)[2:])[::-1], 2)
print([BCR(n) for n in range(75)]) # Michael S. Branicky, Jun 14 2021

def A036044(n): return -int((s:=bin(n)[-1:1:-1]),2)-1+2**len(s) # Chai Wah Wu, Feb 04 2022

a(2n) = 2*A059894(n), a(2n+1) = a(2n) - 2^floor(log_2(n)+1). - Ralf Stephan, Aug 21 2003

Conjecture: a(n) = (-1)^A023416(n)*b(n) for n > 0 with a(0) = 1 where b(2^m) = (-1)^m*(2^(m+1) - 2) for m >= 0, b(2n+1) = b(n) for n > 0, b(2n) = b(n) + b(n - 2^f(n)) + b(2n - 2^f(n)) for n > 0 and where f(n) = A007814(n) (see A329369). - Mikhail Kurkov, Dec 13 2024

cp's OEIS Frontend

A351172 Natural numbers that can be written as the quotient of two antipalindromic numbers (A035928).

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A278465 Numbers that are the sum of one or two terms of A035928.

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A318569 a(n) is the smallest positive integer such that n*a(n) is a "binary antipalindrome" (i.e., an element of A035928).

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A342582 a(n) is the least multiple of n that is a "binary antipalindrome" (i.e., an element of A035928).

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A351173 Natural numbers that cannot be represented as the quotient of two antipalindromic numbers (A035928).

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A351175 Natural numbers k such that there are infinitely many representations k = A/B with A, B in A035928.

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A278546 Even numbers that cannot be expressed as a sum of 3 or fewer terms of A035928.

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Mathematica

A351325 Numbers k with exactly one solution to the equation A = B*k, where A and B are antipalindromic numbers (members of A035928).

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A000740 Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement; also Dirichlet convolution of b_n=2^(n-1) with mu(n); also number of components of Mandelbrot set corresponding to Julia sets with an attractive n-cycle.

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Maple

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PARI