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A010052
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Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.
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340
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1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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0,1
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COMMENTS
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Also parity of the divisor function A000005 if n >= 1. - Omar E. Pol, Jan 14 2012
This sequence can be considered as k=1 analog of A025426 (k=2), A025427 (k=3), A025428 (k=4); see also A000161. - M. F. Hasler, Jan 25 2013
Also, the decimal expansion of Sum_{n >= 0} 1/(10^n)^n. - Eric Desbiaux, Mar 15 2009, rephrased and simplified by M. F. Hasler, Jan 26 2013
Run lengths of zeros gives A005843, the nonnegative even numbers. - Jeremy Gardiner, Jan 14 2018
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 3-4, also p. 166, Ex. 5.5.1.
T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, Problem 20.
M. D. Hirschhorn, The Power of q, Springer, 2017. See phi(q) page 8.
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
D. Christopher and T. Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), #15.11.5.
Robert Price, Comments on A010052 concerning Elementary Cellular Automata, Jan 29 2016
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index entries for characteristic functions
Index to Elementary Cellular Automata
Index entries for sequences computed from exponents in factorization of n
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FORMULA
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a(n) = floor(sqrt(n)) - floor(sqrt(n-1)), for n > 0.
a(n) = A000005(n) mod 2, n > 0. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
G.f. A(x) satisfies: 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-w)^2 - (v-w)*(v+w-1) - Michael Somos, Jul 19 2004
Dirichlet g.f.: zeta(2s). - Franklin T. Adams-Watters, Sep 11 2005
G.f.: (theta_3(0,x) + 1)/2, where theta_3 is a Jacobi theta function. - Franklin T. Adams-Watters, Jun 19 2006 [See A000122 for theta_3.]
a(n) = f(n,0) with f(x,y) = f(x-2*y-1,y+1) if x > 0, otherwise 0^(-x). - Reinhard Zumkeller, Sep 26 2008
a(n) = sumdiv(n,d,(-1)^bigomega(d)), for n >= 1. - Benoit Cloitre, Oct 25 2009
a(n) <= A093709(n). - Reinhard Zumkeller, Nov 14 2009
a(A000290(n)) = 1; a(A000037(n)) = 0. - Reinhard Zumkeller, Jun 20 2011
a(n) = 0 ^ A053186(n). - Reinhard Zumkeller, Feb 12 2012
a(n) = A063524(A007913(n)), for n > 0. - Reinhard Zumkeller, Jul 09 2014
a(n) = -(-1)^n * A258998(n) unless n = 0. 2 * a(n) = A000122(n) unless n = 0. - Michael Somos, Jun 16 2015
a(n) = A037011(A156552(n)), provided that A037011(n) = A000035(A106737(n)). [See A037011.] - Antti Karttunen, Nov 03 2017
a(n*m) = a(n/gcd(n,m))*a(m/gcd(n,m)) for all n and m > 0 (conjectured). - Velin Yanev, Feb 13 2019 [Proof from Michael B. Porter, Feb 16 2019: If nm is a square, nm = product_i (p_i^2), where p_i are prime, not necessarily distinct. Each p_i either appears twice in n, twice in m, or one time in each and therefore in the gcd. So n/gcd(n,m) and m/gcd(n,m) are both squares. If nm is not a square, there is a q_j that appears in one of n or m but not in the gcd. So either n/gcd(n,m) or m/gcd(n,m) is not a square.]
a(n) = Sum_{ d | n } A008836(d). - Jinyuan Wang, Apr 20 2019
G.f.: A(q) = Sum_{n >= 0} q^(2*n)*Product_{k >= 2*n+1} 1 - (-q)^k. - Peter Bala, Feb 22 2021
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EXAMPLE
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G.f. = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 + x^81 + ...
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MAPLE
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readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i), i=0..100) ];
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MATHEMATICA
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lst = {}; Do[AppendTo[lst, 2*Sum[Floor[n/k] - Floor[(n - 1)/k], {k, Floor[Sqrt[n]]}] - DivisorSigma[0, n]], {n, 93}]; Prepend[lst, 1] (* Eric Desbiaux, Jan 29 2012 *)
Table[If[IntegerQ[Sqrt[n]], 1, 0], {n, 0, 100}] (* Harvey P. Dale, Jul 19 2014 *)
a[n_] := SeriesCoefficient[1/(1 - q)* QHypergeometricPFQ[{-q, -q}, {-(q^2)}, -q, -q], {q, 0, Abs@n}] (* Mats Granvik, Jan 01 2016 *)
Range[0, 120] /. {n_ /; IntegerQ@ Sqrt@ n -> 1, n_ /; n != 1 -> 0} (* Michael De Vlieger, Jan 02 2016 *)
a[n_] := Sum[If[Mod[n, k] == 0, Re[Sqrt[LiouvilleLambda[k]]*Sqrt[LiouvilleLambda[n/k]]], 0], {k, 1, n}] (* Mats Granvik, Aug 10 2018 *)
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PROG
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(PARI) {a(n) = issquare(n)};
(PARI) a(n)=if(n<1, 1, sumdiv(n, d, (-1)^bigomega(d))) \\ Benoit Cloitre, Oct 25 2009
(PARI) a(n) = if (n<1, 1, direuler( p=2, n, 1/ (1 - X^2 ))[n]); \\ Michel Marcus, Mar 08 2015
(Haskell)
a010052 n = fromEnum $ a000196 n ^ 2 == n
-- Reinhard Zumkeller, Jan 26 2012, Feb 20 2011
a010052_list = concat (iterate (\xs -> xs ++ [0, 0]) [1])
-- Reinhard Zumkeller, Apr 27 2012
(Scheme) (define (A010052 n) (if (zero? n) 1 (- (A000196 n) (A000196 (- n 1))))) ;; (For the definition of A000196, see under that entry). - Antti Karttunen, Nov 03 2017
(Python)
def A010052(n): return int(math.isqrt(n)**2==n) ## appears to be faster than sympy.ntheory.primetest.is_square, up to 10^8 at least.
# M. F. Hasler, Mar 21 2022
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CROSSREFS
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Column k=1 of A243148, A337165, A341040 (for n>0).
Cf. A000005, A000122, A005369, A007913, A008836, A037011, A063524, A258998, A271102 (Dirichlet inv).
First differences of A000196.
Sequence in context: A127692 A014305 A023533 * A302052 A039985 A324822
Adjacent sequences: A010049 A010050 A010051 * A010053 A010054 A010055
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KEYWORD
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nonn,nice,easy,mult
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Franklin T. Adams-Watters, Jun 19 2006
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STATUS
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approved
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