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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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362
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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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Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 3-4, also p. 166, Ex. 5.5.1.
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, Problem 20.
Michael 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.
Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 55.
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LINKS
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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*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
Multiplicative with a(p^e) = 1 if e is even, and 0 otherwise. - _Amiram Eldar_, Dec 29 2022
a(n) = Sum_{d divides n} mobius(core(n)), where core(n) = A007913(n). - _Peter Bala_, Jan 24 2024
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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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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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