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A010052 Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0. 361
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Christopher and Tamil Nadu, Partitions with Fixed Number of Sizes, Journal of Integer Sequences, 15 (2015), Article 15.11.5.
Yash Puri and Thomas Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), Article 01.2.1.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
Stephen Wolfram, A New Kind of Science.
FORMULA
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
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
EXAMPLE
G.f. = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + x^64 + x^81 + ...
MAPLE
readlib(issqr): f := i->if issqr(i) then 1 else 0; fi; [ seq(f(i), i=0..100) ];
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Column k=1 of A243148, A337165, A341040 (for n>0).
Cf. A000005, A000122, A005369, A007913, A008836 (Mobius transf.), A037011, A063524, A258998, A271102 (Dirichlet inv), A046951 (inv. Mobius trans.).
First differences of A000196.
Sequence in context: A127692 A014305 A023533 * A302052 A039985 A324822
KEYWORD
nonn,nice,easy,mult
AUTHOR
EXTENSIONS
More terms from Franklin T. Adams-Watters, Jun 19 2006
STATUS
approved

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Last modified February 28 05:20 EST 2024. Contains 370379 sequences. (Running on oeis4.)