A019568 -id:A019568 - cp's OEIS frontend

A000069 Odious numbers: numbers with an odd number of 1's in their binary expansion.

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 64, 67, 69, 70, 73, 74, 76, 79, 81, 82, 84, 87, 88, 91, 93, 94, 97, 98, 100, 103, 104, 107, 109, 110, 112, 115, 117, 118, 121, 122, 124, 127, 128

Offset: 1

N. J. A. Sloane

This sequence and A001969 give the unique solution to the problem of splitting the nonnegative integers into two classes in such a way that sums of pairs of distinct elements from either class occur with the same multiplicities [Lambek and Moser]. Cf. A000028, A000379.
In French: les nombres impies.
Has asymptotic density 1/2, since exactly 2 of the 4 numbers 4k, 4k+1, 4k+2, 4k+3 have an even sum of bits, while the other 2 have an odd sum. - Jeffrey Shallit, Jun 04 2002
Nim-values for game of mock turtles played with n coins.
A115384(n) = number of odious numbers <= n; A000120(a(n)) = A132680(n). - Reinhard Zumkeller, Aug 26 2007
Indices of 1's in the Thue-Morse sequence A010060. - Tanya Khovanova, Dec 29 2008
For any positive integer m, the partition of the set of the first 2^m positive integers into evil ones E and odious ones O is a fair division for any polynomial sequence p(k) of degree less than m, that is, Sum_{k in E} p(k) = Sum_{k in O} p(k) holds for any polynomial p with deg(p) < m. - Pietro Majer, Mar 15 2009
For n>1 let b(n) = a(n-1). Then b(b(n)) = 2b(n). - Benoit Cloitre, Oct 07 2010
Lexicographically earliest sequence of distinct nonnegative integers with no term being the binary exclusive OR of any terms. The equivalent sequence for addition or for subtraction is A005408 (the odd numbers) and for multiplication is A026424. - Peter Munn, Jan 14 2018
Numbers of the form m XOR (2*m+1) for some m >= 0. - Rémy Sigrist, Apr 14 2022

			For k=2, x=0 and x=0.2 we respectively have 1^2 + 2^2 + 4^2 + 7^2 = 0^2 + 3^2 + 5^2 + 6^2 = 70;
(1.2)^2 + (2.2)^2 + (4.2)^2 + (7.2)^2 = (0.2)^2 + (3.2)^2 + (5.2)^2 + (6.2)^2 = 75.76;
for k=3, x=1.8, we have (2.8)^3 + (3.8)^3 + (5.8)^3 + (8.8)^3 + (9.8)^3 + (12.8)^3 + (14.8)^3 + (15.8)^3 = (1.8)^3 + (4.8)^3 + (6.8)^3 + (7.8)^3 + (10.8)^3 + (11.8)^3 + (13.8)^3 + (16.8)^3 = 11177.856. - _Vladimir Shevelev_, Jan 16 2012

E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, NY, 2 vols., 1982, see p. 433.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 22.
Vladimir S. Shevelev, On some identities connected with the partition of the positive integers with respect to the Morse sequence, Izv. Vuzov of the North-Caucasus region, Nature sciences 4 (1997), 21-23 (in Russian).
N. J. A. Sloane, A handbook of Integer Sequences, Academic Press, 1973 (including this sequence).
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 = 1..10001
Jean-Paul Allouche, The zeta-regularized product of odious numbers, arXiv:1906.10532 [math.NT], 2019.
Jean-Paul Allouche and Jeffrey Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
Jean-Paul Allouche, Jeffrey Shallit and G. Skordev, Self-generating sets, integers with missing blocks and substitutions, Discrete Math. 292 (2005) 1-15.
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, Aequationes mathematicae, March 2015, pp 1-13.
Jean-Paul Allouche, Jeffrey Shallit, and Manon Stipulanti, Combinatorics on words and generating Dirichlet series of automatic sequences, arXiv:2401.13524 [math.CO], 2025. See p. 19.
E. Fouvry and C. Mauduit, Sommes des chiffres et nombres presque premiers, (French) [Sums of digits and almost primes] Math. Ann. Vol. 305, No. 1 (1996), 571-599, DOI:10.1007/BF01444238, MR1397437 (97k:11029).
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Mathematics, Volume 312, Issue 1, 6 January 2012, Pages 42-46.
Maciej Gawron, and Maciej Ulas, On formal inverse of the Prouhet-Thue-Morse sequence, Discrete Mathematics 339.5 (2016): 1459-1470. Also arXiv preprint, arXiv:1601.04840 [math.CO], 2016.
R. K. Guy, The unity of combinatorics, Proc. 25th Iranian Math. Conf, Tehran, (1994), Math. Appl 329 129-159, Kluwer Dordrecht 1995, Math. Rev. 96k:05001.
R. K. Guy, Impartial games, pp. 35-55 of Combinatorial Games, ed. R. K. Guy, Proc. Sympos. Appl. Math., 43, Amer. Math. Soc., 1991.
Sajed Haque, Chapter 3.2 of Discriminators of Integer Sequences, Thesis, 2017.
Sajed Haque and Jeffrey Shallit, Discriminators and k-Regular Sequences, arXiv:1605.00092 [cs.DM], 2016.
K. Jensen, Aesthetics and quality of numbers using the primety measure, Int. J. Arts and Technology, Vol. 7, Nos. 2/3, 2014.
Clark Kimberling, Affinely recursive sets and orderings of languages, Discrete Math., 274 (2004), 147-160. [From _N. J. A. Sloane_, Jan 31 2012]
Tanya Khovanova, There are no coincidences, arXiv:1410.2193 [math.CO], 2014.
J. Lambek and L. Moser, On some two way classifications of integers, Canad. Math. Bull. 2 (1959), 85-89.
M. D. McIlroy, The number of 1's in binary integers: bounds and extremal properties, SIAM J. Comput., 3 (1974), 255-261.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 208.
D. J. Newman, A Problem Seminar, Problem 15 pp. 5; 15 Springer-Verlag NY 1982.
Aayush Rajasekaran, Jeffrey Shallit and Tim Smith, Additive Number Theory via Automata Theory, Theory of Computing Systems (2019) 1-26.
Jeffrey Shallit, Additive Number Theory via Automata and Logic, arXiv:2112.13627 [math.NT], 2021.
Vladimir Shevelev and Peter J. C. Moses, Tangent power sums and their applications, arXiv:1207.0404 [math.NT], 2012-2014. - From _N. J. A. Sloane_, Dec 17 2012
Vladimir Shevelev and Peter J. C. Moses, Tangent power sums and their applications, INTEGERS, 14(2014) #64.
Vladimir Shevelev and Peter J. C. Moses, A family of digit functions with large periods, arXiv:1209.5705 [math.NT], 2012.
Andrzej Tomski and Maciej Zakarczemny, A note on Browkin's and Cao's cancellation algorithm, Technical Transections 7/2018.
Eric Weisstein's World of Mathematics, Odious Number
Index entries for sequences related to binary expansion of n
Index entries for "core" sequences

The basic sequences concerning the binary expansion of n are A000120, A000788, A000069, A001969, A023416, A059015.
Complement of A001969 (the evil numbers). Cf. A133009.
a(n) = 2*n + 1 - A010060(n) = A001969(n) + (-1)^A010060(n).
First differences give A007413.
Cf. A000773, A181155, A019568, A059009.
Note that A000079, A083420, A002042, A002089, A132679 are subsequences.
See A027697 for primes, also A230095.
Cf. A005408 (odd numbers), A006068, A026424.
Cf. A010059, A010060.

a000069 n = a000069_list !! (n-1)
a000069_list = [x | x <- [0..], odd $ a000120 x]
-- Reinhard Zumkeller, Feb 01 2012

[ n: n in [1..130] | IsOdd(&+Intseq(n, 2)) ]; // Klaus Brockhaus, Oct 07 2010

s := proc(n) local i,j,k,b,sum,ans; ans := [ ]; j := 0; for i while jA000069 := n->t1[n]; # s(k) gives first k terms.
is_A000069 := n -> type(add(i,i=convert(n,base,2)),odd):
seq(`if`(is_A000069(i),i,NULL),i=0..40); # Peter Luschny, Feb 03 2011

Select[Range[300], OddQ[DigitCount[ #, 2][[1]]] &] (* Stefan Steinerberger, Mar 31 2006 *)
a[ n_] := If[ n < 1, 0, 2 n - 1 - Mod[ Total @ IntegerDigits[ n - 1, 2], 2]]; (* Michael Somos, Jun 01 2013 *)

{a(n) = if( n<1, 0, 2*n - 1 - subst( Pol(binary( n-1)), x, 1) % 2)}; /* Michael Somos, Jun 01 2013 */

{a(n) = if( n<2, n==1, if( n%2, a((n+1)/2) + n-1, -a(n/2) + 3*(n-1)))}; /* Michael Somos, Jun 01 2013 */

a(n)=2*n-1-hammingweight(n-1)%2 \\ Charles R Greathouse IV, Mar 22 2013

[n for n in range(1, 201) if bin(n)[2:].count("1") % 2] # Indranil Ghosh, May 03 2017

def A000069(n): return ((m:=n-1)<<1)+(m.bit_count()&1^1) # Chai Wah Wu, Mar 03 2023

G.f.: 1 + Sum_{k>=0} (t*(2+2t+5t^2-t^4)/(1-t^2)^2) * Product_{j=0..k-1} (1-x^(2^j)), t=x^2^k. - Ralf Stephan, Mar 25 2004
a(n+1) = (1/2) * (4*n + 1 + (-1)^A000120(n)). - Ralf Stephan, Sep 14 2003
Numbers n such that A010060(n) = 1. - Benoit Cloitre, Nov 15 2003
a(2*n+1) + a(2*n) = A017101(n) = 8*n+3. a(2*n+1) - a(2*n) gives the Thue-Morse sequence (1, 3 version): 1, 3, 3, 1, 3, 1, 1, 3, 3, 1, 1, 3, 1, ... A001969(n) + A000069(n) = A016813(n) = 4*n+1. - Philippe Deléham, Feb 04 2004
(-1)^a(n) = 2*A010060(n)-1. - Benoit Cloitre, Mar 08 2004
a(1) = 1; for n > 1: a(2*n) = 6*n-3 -a(n), a(2*n+1) = a(n+1) + 2*n. - Corrected by Vladimir Shevelev, Sep 25 2011
For k >= 1 and for every real (or complex) x, we have Sum_{i=1..2^k} (a(i)+x)^s = Sum_{i=1..2^k} (A001969(i)+x)^s, s=0..k.
For x=0, s <= k-1, this is known as Prouhet theorem (see J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence). - Vladimir Shevelev, Jan 16 2012
a(n+1) mod 2 = 1 - A010060(n) = A010059(n). - Robert G. Wilson v, Jan 18 2012
A005590(a(n)) > 0. - Reinhard Zumkeller, Apr 11 2012
A106400(a(n)) = -1. - Reinhard Zumkeller, Apr 29 2012
a(n+1) = A006068(n) XOR (2*A006068(n) + 1). - Rémy Sigrist, Apr 14 2022

A063865 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 8, 14, 0, 0, 70, 124, 0, 0, 722, 1314, 0, 0, 8220, 15272, 0, 0, 99820, 187692, 0, 0, 1265204, 2399784, 0, 0, 16547220, 31592878, 0, 0, 221653776, 425363952, 0, 0, 3025553180, 5830034720, 0, 0, 41931984034, 81072032060, 0, 0

Offset: 0

N. J. A. Sloane, suggested by J. H. Conway, Aug 27 2001

Number of sum partitions of the half of the n-th-triangular number by distinct numbers in the range 1 to n. Example: a(7)=8 since triangular(7)=28 and 14 = 2+3+4+5 = 1+3+4+6 = 1+2+5+6 = 3+5+6 = 7+1+2+4 = 7+3+4 = 7+2+5 = 7+1+6. - Hieronymus Fischer, Oct 20 2010
The asymptotic formula below was stated as a conjecture by Andrica & Tomescu in 2002 and proved by B. D. Sullivan in 2013. See his paper and H.-K. Hwang's review MR 2003j:05005 of the JIS paper. - Jonathan Sondow, Nov 11 2013
a(n) is the number of subsets of {1..n} whose sum is equal to the sum of their complement. See example below. - Gus Wiseman, Jul 04 2019

			From _Gus Wiseman_, Jul 04 2019: (Start)
For example, the a(0) = 1 through a(8) = 14 subsets (empty columns not shown) are:
  {}  {3}    {1,4}  {1,6,7}    {3,7,8}
      {1,2}  {2,3}  {2,5,7}    {4,6,8}
                    {3,4,7}    {5,6,7}
                    {3,5,6}    {1,2,7,8}
                    {1,2,4,7}  {1,3,6,8}
                    {1,2,5,6}  {1,4,5,8}
                    {1,3,4,6}  {1,4,6,7}
                    {2,3,4,5}  {2,3,5,8}
                               {2,3,6,7}
                               {2,4,5,7}
                               {3,4,5,6}
                               {1,2,3,4,8}
                               {1,2,3,5,7}
                               {1,2,4,5,6}
(End)

T. D. Noe, N. J. A. Sloane and Ray Chandler, Table of n, a(n) for n = 0..3339 (terms < 10^1000, first 101 terms from T. D. Noe, next 300 terms from N. J. A. Sloane)
Dorin Andrica and Ovidiu Bagdasar, On k-partitions of multisets with equal sums, The Ramanujan J. (2021) Vol. 55, 421-435.
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Ţurcaş, The Number of Partitions of a Set and Superelliptic Diophantine Equations, Disc. Math. and Applications, Springer, Cham (2020), 35-55.
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
D. Andrica and I. Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Seq., 5 (2002), Article 02.2.4
Ovidiu Bagdasar and Dorin Andrica, New results and conjectures on 2-partitions of multisets, 2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO).
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]
B. D. Sullivan, On a Conjecture of Andrica and Tomescu, J. Int. Sequences, 16 (2013), Article 13.3.1.
zbMATH, Review of Andrica and Tomescu

Cf. A000980, A025591, A058377, A063866, A063867, A113036, A113037, A141000.
"Decimations": A060468 = 2*A060005, A123117 = 2*A104456.
Analogous sequences for sums of squares and cubes are A158092, A158118, see also A019568. - Pietro Majer, Mar 15 2009
Cf. A053632, A059529, A326173, A326174, A326175.

M:=400; t1:=1; lprint(0,1); for n from 1 to M do t1:=expand(t1*(x^n+1/x^n)); lprint(n, coeff(t1,x,0)); od: # N. J. A. Sloane, Jul 07 2008

f[n_, s_] := f[n, s]=Which[n==0, If[s==0, 1, 0], Abs[s]>(n*(n+1))/2, 0, True, f[ n-1, s-n]+f[n-1, s+n]]; a[n_] := f[n, 0]
nmax = 50; d = {1}; a1 = {};
Do[
  i = Ceiling[Length[d]/2];
  AppendTo[a1, If[i > Length[d], 0, d[[i]]]];
  d = PadLeft[d, Length[d] + 2 n] + PadRight[d, Length[d] + 2 n];
  , {n, nmax}];
a1 (* Ray Chandler, Mar 13 2014 *)

a(n)=my(x='x); polcoeff(prod(k=1,n,x^k+x^-k)+O(x),0) \\ Charles R Greathouse IV, May 18 2015

a(n)=0^n+floor(prod(k=1,n,2^(n*k)+2^(-n*k)))%(2^n) \\ Tani Akinari, Mar 09 2016

Asymptotic formula: a(n) ~ sqrt(6/Pi)*n^(-3/2)*2^n for n = 0 or 3 (mod 4) as n approaches infinity.
a(n) = 0 unless n == 0 or 3 (mod 4).
a(n) = constant term in expansion of Product_{ k = 1..n } (x^k + 1/x^k). - N. J. A. Sloane, Jul 07 2008
If n = 0 or 3 (mod 4) then a(n) = coefficient of x^(n(n+1)/4) in Product_{k=1..n} (1+x^k). - D. Andrica and I. Tomescu.
a(n) = 2*A058377(n) for any n > 0. - Rémy Sigrist, Oct 11 2017

More terms from Dean Hickerson, Aug 28 2001

Corrected and edited by Steven Finch, Feb 01 2009

A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 10, 0, 0, 86, 114, 0, 0, 478, 860, 0, 0, 5808, 10838, 0, 0, 55626, 100426, 0, 0, 696164, 1298600, 0, 0, 7826992, 14574366, 0, 0, 100061106, 187392994, 0, 0, 1223587084, 2322159814, 0, 0, 16019866270, 30353305134, 0, 0

Offset: 1

Pietro Majer, Mar 12 2009

nonn

Twice A083527.
Number of partitions of the half of the n-th-square-pyramidal number into parts that are distinct square numbers in the range 1 to n^2. Example: a(7)=2 since, squarePyramidal(7)=140 and 70=1+4+16+49=9+25+36. - Hieronymus Fischer, Oct 20 2010
Erdős & Surányi prove that this sequence is unbounded. More generally, there are infinitely many ways to write a given number k as such a sum. - Charles R Greathouse IV, Nov 05 2012
The expansion and integral representation formulas below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - Jonathan Sondow, Nov 11 2013

			For n=8 the a(8)=2 solutions are: +1-4-9+16-25+36+49-64=0 and -1+4+9-16+25-36-49+64=0.

Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..500 (first 240 terms from Alois P. Heinz)
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.
P. Erdős and J. Surányi, Egy additív számelméleti probléma (in Hungarian; Russian and German summaries), Mat. Lapok 10 (1959), pp. 284-290.

Cf. A063865, A158118, A019568, A083527, A231015.

From Pietro Majer, Mar 15 2009: (Start)
N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*(x^(n^2)+x^(-n^2))):
a:=[op(a), coeff(p, x, 0)]: od:a; (End)
# second Maple program:
b:= proc(n, i) option remember; local m; m:= (1+(3+2*i)*i)*i/6;
      `if`(n>m, 0, `if`(n=m, 1, b(abs(n-i^2), i-1) +b(n+i^2, i-1)))
    end:
a:= n-> `if`(irem(n-1, 4)<2, 0, 2*b(n^2, n-1)):
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 05 2012

b[n_, i_] := b[n, i] = With[{m = (1+(3+2*i)*i)*i/6}, If[n>m, 0, If[n == m, 1, b[ Abs[n-i^2], i-1] + b[n+i^2, i-1]]]]; a[n_] := If[Mod[n-1, 4]<2, 0, 2*b[n^2, n-1]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(n)=2*sum(i=0,2^(n-1)-1,sum(j=1,n-1,(-1)^bittest(i,j-1)*j^2)==n^2) \\ Charles R Greathouse IV, Nov 05 2012

from itertools import count, islice
from collections import Counter
def A158092_gen(): # generator of terms
    ccount = Counter({0:1})
    for i in count(1):
        bcount = Counter()
        for a in ccount:
            bcount[a+(j:=i**2)] += ccount[a]
            bcount[a-j] += ccount[a]
        ccount = bcount
        yield(ccount[0])
A158092_list = list(islice(A158092_gen(),20)) # Chai Wah Wu, Jan 29 2024

Constant term in the expansion of (x + 1/x)(x^4 + 1/x^4)..(x^n^2 + 1/x^n^2).
a(n)=0 for any n == 1 or 2 (mod 4).
Integral representation: a(n)=((2^n)/pi)*int_0^pi prod_{k=1}^n cos(x*k^2) dx
Asymptotic formula: a(n) = (2^n)*sqrt(10/(pi*n^5))*(1+o(1)) as n-->infty; n == -1 or 0 (mod 4).
a(n) = 2 * A083527(n). - T. D. Noe, Mar 12 2009
min{n : a(n) > 0} = A231015(0) = 7. - Jonathan Sondow, Nov 06 2013

a(51)-a(56) from R. H. Hardin, Mar 12 2009

Edited by N. J. A. Sloane, Sep 15 2009

A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 2, 0, 0, 4, 124, 0, 0, 536, 712, 0, 0, 4574, 2260, 0, 0, 10634, 73758, 0, 0, 406032, 638830, 0, 0, 4249160, 3263500, 0, 0, 21907736, 82561050, 0, 0, 485798436, 945916970, 0, 0, 5968541478, 6839493576, 0, 0

Offset: 1

Pietro Majer, Mar 12 2009

nonn

Constant term in the expansion of (x + 1/x)(x^8 + 1/x^8)..(x^n^3 + 1/x^n^3).
a(n) = 0 for any n=1 (mod 4) or n=2 (mod 4).
The expansion above and the integral representation formula below are due to Andrica & Tomescu. The asymptotic formula is a conjecture; see Andrica & Ionascu. - Jonathan Sondow, Nov 06 2013

			Example: For n=12 the a(12) = 2 solutions are:
+1+8-27+64-125-216-343+512+729-1000-1331+1728=0,
-1-8+27-64+125+216+343-512-729+1000+1331-1728=0.

Ray Chandler, Table of n, a(n) for n = 1..130
D. Andrica and E. J. Ionascu, Variations on a result of Erdős and Surányi, INTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Equals twice A113263.

Cf. A063865, A158092, A019568. - Pietro Majer, Mar 15 2009

N:=60: p:=1: a:=[]: for n from 1 to N do p:=expand(p*( x^(n^3) + x^(-n^3) )): a:=[op(a), coeff(p,x,0)]: od:a;

a(n) = 2 * A113263(n).
Integral representation: a(n)=((2^n)/Pi)*int_0^Pi prod_{k=1}^n cos(x*k^3) dx.
Asymptotic formula: a(n)=(2^n)*sqrt(14/(Pi*n^7))*(1+o(1)) as n-->infty; n=-1 or 0 (mod 4).

A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 16, 18, 0, 0, 32, 100, 0, 0, 424, 510, 0, 0, 2792, 5988, 0, 0, 29058, 45106, 0, 0, 276828, 473854, 0, 0, 2455340, 4777436, 0, 0, 27466324, 46429640, 0, 0, 280395282, 526489336, 0, 0, 3193589950, 5661226928, 0, 0

Offset: 1

Pietro Majer, Mar 19 2009

nonn

Constant term in the expansion of (x + 1/x)(x^16 + 1/x^16)..(x^n^4 + 1/x^n^4).
a(n)=0 for any n=1 (mod 4) or n=2 (mod 4).
Andrica & Tomescu give a more general integral formula than the one below. The asymptotic formula below is a conjecture by Andrica & Ionascu; it remains unproven. - Jonathan Sondow, Nov 11 2013

			For n=16 the a(16) = 2 solutions are +1 +16 +81 +256 -625 -1296 -2401 +4096 +6561 +10000 +14641 +20736 -28561 -38416 -50625 +65536 = 0 and the opposite.

D. Andrica and E. J. Ionascu, Variations on a result of Erdős and SurányiINTEGERS 2013 slides.
Dorin Andrica and Ioan Tomescu, On an Integer Sequence Related to a Product of Trigonometric Functions, and Its Combinatorial Relevance, J. Integer Sequences, 5 (2002), Article 02.2.4.

Cf. A063865, A158092, A158118, A158380, A019568.

A111253(n) = a(n)/2. - Alois P. Heinz, Oct 31 2011

N:=32: p:=1 a:=[]: for n from 32 to N do p:=expand
(p*(x^(n^4)+x^(-n^4))): a:=[op(a), coeff(p,x,0)]: od:a;

Integral representation: a(n) = ((2^n)/Pi)*int_0^pi prod_{k=1}^n cos(x*k^4) dx.

Asymptotic formula: a(n) = (2^n)*sqrt(18/(Pi*n^9))*(1+o(1)) as n->infinity; n=-1 or 0 (mod 4).

a(35)-a(58) from Alois P. Heinz, Oct 31 2011

A240070 a(n) is the least k such that {1^n, 2^n, 3^n, ..., k^n} can be partitioned into 3 sets whose sums are equal.

Original entry on oeis.org

3, 5, 13, 23, 35, 30, 45, 53, 71, 71

Offset: 0

T. D. Noe, Apr 02 2014

			For n = 0, the 3 numbers are partitioned as {1}, {2}, {3}.
For n = 1, the 5 numbers are partitioned as {1,4}, {2,3}, {5}.
For n = 2, the 13 numbers are partitioned as {2,10,13}, {4,7,8,12}, {1,3,5,6,9,11}.
For n = 3, the 23 numbers are partitioned as {3,6,10,13,18,19,21}, {1,4,7,8,12,16,20,22}, {2,5,9,11,14,15,17,23}.
From _Alois P. Heinz_, Apr 04 2014: (Start)
One partition for n=4 is {1,2,12,17,18,20,21,22,23,24,25,27,28,30}, {3,4,5,6,9,13,14,16,26,31,32,33}, {7,8,10,11,15,19,29,34,35}.
One partition for n=5 is {1,2,3,6,13,14,15,17,19,20,22,23,30}, {4,7,8,10,12,16,18,25,27,28}, {5,9,11,21,24,26,29}. (End)
One partition for n=6 is {1,3,5,9,11,14,15,19,20,21,27,29,34,35,43,45}, {4,6,7,10,12,13,16,17,24,28,33,36,38,41,44}, {2,8,18,22,23,25,26,30,31,32,37,39,40,42}.
One partition for n=7 is {2,4,5,6,8,12,16,20,21,22,25,27,29,30,32,35,36,40,46,50,53}, {1,3,10,13,14,18,24,28,33,34,37,38,39,41,42,45,47,52}, {7,9,11,15,17,19,23,26,31,43,44,48,49,51}.
From _Michael S. Branicky_, Jul 02 2020: (Start)
One partition for n=8 is {1,2,5,7,11,13,18,24,25,26,27,29,32,34,35,36,41,52,66,67,68,69}, {4,8,9,15,17,20,21,23,30,43,47,50,53,54,55,57,60,61,62,63,70}, {3,6,10,12,14,16,19,22,28,31,33,37,38,39,40,42,44,45,46,48,49,51,56,58,59,64,65,71}.
One partition for n=9 is
  {2,4,5,6,10,12,13,19,20,35,37,44,50,61,63,70,71},
  {1,7,22,24,26,27,29,31,33,36,38,39,40,41,42,46,47,49,52,54,56,59,66,68,69}, {3,8,9,11,14,15,16,17,18,21,23,25,28,30,32,34,43,45,48,51,53,55,57,58,60,62,64,65,67}. (End)

Cf. A019568 (partitioned into 2 sets).

goodQ[set_, n_] :=
Module[{found = False},
  Do[If[Union[set[[i]], set[[j]], set[[k]]] == Range[n] &&
     Length[set[[i]]] + Length[set[[j]]] + Length[set[[k]]] == n,
    Print[{set[[i]], set[[j]], set[[k]]}]; found = True; Break[]], {i,
     Length[set]}, {j, i, Length[set]}, {k, j, Length[set]}]; found];
Join[{3}, Table[k = 1; found = False;
While[s = Range[k]^n; sm = Total[s]/3;
  If[IntegerQ[sm],
   t3 = Select[Subsets[s], Total[#] == sm &]^(1/n);
   found = goodQ[t3, k]]; ! found, k++]; k, {n, 3}]]

a(4)-a(5) from Alois P. Heinz, Apr 03 2014
a(6)-a(7) from Dean D. Ballard, Jun 09 2020
a(8)-a(9) from Michael S. Branicky, Jul 02 2020

A173209 Partial sums of A000069.

Original entry on oeis.org

1, 3, 7, 14, 22, 33, 46, 60, 76, 95, 116, 138, 163, 189, 217, 248, 280, 315, 352, 390, 431, 473, 517, 564, 613, 663, 715, 770, 826, 885, 946, 1008, 1072, 1139, 1208, 1278, 1351, 1425, 1501, 1580, 1661, 1743, 1827, 1914, 2002, 2093, 2186, 2280, 2377, 2475, 2575

Offset: 1

Jonathan Vos Post, Feb 12 2010

Partial sums of odious numbers. Second differences give A007413. The subsequence of prime partial sums of odious numbers begins: 3, 7, 163, 431, 613, 2377, 3691, which is a subsequence of A027697. The subsequence of odious partial sums of odious numbers begins: 1, 7, 14, 22, 76, 138, 217, 280, 352, 431, 517, 613, 770, 885.

			a(65) = 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 19 + 21 + 22 + 25 + 26 + 28 + 31 + 32 + 35 + 37 + 38 + 41 + 42 + 44 + 47 + 49 + 50 + 52 + 55 + 56 + 59 + 61 + 62 + 64 + 67 + 69 + 70 + 73 + 74 + 76 + 79 + 81 + 82 + 84 + 87 + 88 + 91 + 93 + 94 + 97 + 98 + 100 + 103 + 104 + 107 + 109 + 110 + 112 + 115 + 117 + 118 + 121 + 122 + 124 + 127 + 128.

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Jean-Paul Allouche, Benoit Cloitre, and Vladimir Shevelev, Beyond odious and evil, arXiv preprint arXiv:1405.6214 [math.NT], 2014.
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. 52.

Cf. A000069, A000079, A000120, A000773, A000788, A001969, A002042, A002089, A007413, A019568, A023416, A027697, A059015, A083420, A010060, A132679, A133009.

Accumulate@ Select[Range[100], OddQ@ DigitCount[#, 2, 1] &] (* Michael De Vlieger, Nov 01 2022 *)

a(n)=n^2-(n+1)\2+(n%2&&hammingweight(n)%2) \\ Charles R Greathouse IV, Mar 22 2013

a(n) = SUM[i=1..n] A000069(i) = SUM[i=1..n] {i such that A010060(i)=1}.

a(n) = n^2 - n/2 + O(1). - Charles R Greathouse IV, Mar 22 2013

cp's OEIS Frontend

A000069 Odious numbers: numbers with an odd number of 1's in their binary expansion.

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A063865 Number of solutions to +- 1 +- 2 +- 3 +- ... +- n = 0.

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A158092 Number of solutions to +- 1 +- 2^2 +- 3^2 +- 4^2 +- ... +- n^2 = 0.

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A158118 Number of solutions of +-1+-2^3+-3^3..+-n^3=0.

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A158465 Number of solutions to +-1+-2^4+-3^4+-4^4...+-n^4=0.

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A240070 a(n) is the least k such that {1^n, 2^n, 3^n, ..., k^n} can be partitioned into 3 sets whose sums are equal.

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