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 A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2. 28
 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005 Characteristic function of numbers coprime to 5. - Reinhard Zumkeller, Nov 30 2009 From R. J. Mathar, Jul 15 2010: (Start) The sequence is the principal Dirichlet character mod 5. (The other real character mod 5 is A080891.) Associated Dirichlet L-functions are for example L(2,chi) = Sum_{n>=1} a(n)/n^2 = 1.5791367... = (psi'(1/5) + psi'(2/5) + psi'(3/5) + psi'(4/5))/25 or L(3,chi) = Sum_{n>=1} a(n)/n^3 = 1.192440... = -(psi''(1/5) + psi''(2/5) + psi''(3/5) + psi''(4/5))/250, where psi' and psi'' are the trigamma and tetragamma functions. (End) a(n) is for n >= 1 also the characteristic function for rational g-adic integers (+n/5)_g and also (-n/5)_g for all integers g >= 2 without a factor of 5 (A047201). See the definition in the Mahler reference, p. 7 and also p. 10. - Wolfdieter Lang, Jul 11 2014 Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - R. J. Mathar, Mar 01 2023 REFERENCES Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10). K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981. LINKS Antti Karttunen, Table of n, a(n) for n = 0..65537 Michael Gilleland, Some Self-Similar Integer Sequences R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985. Index entries for characteristic functions Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1). FORMULA O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - Wolfdieter Lang, Feb 05 2009 From Reinhard Zumkeller, Nov 30 2009: (Start) a(n) = 1 - A079998(n). a(A047201(n))=1, a(A008587(n))=0. A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End) a(n) = n^4 mod 5. - Gary Detlefs, Mar 20 2010 Sum_{n>=1} a(n)/n^s = L(s,chi) = (1-1/5^s)*Riemann_zeta(s), s > 1. - R. J. Mathar, Jul 31 2010 For the general case. The characteristic function of numbers that are not multiples of m is a(n) = floor((n-1)/m) - floor(n/m) + 1, m,n > 0. - Boris Putievskiy, May 08 2013 a(n) = sgn(n mod 5). - Wesley Ivan Hurt, Jun 30 2013 Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - Michael Somos, May 24 2015 Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - Michael Somos, May 24 2015 G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - Michael Somos, May 24 2015 |a(n)| = |A080891(n)| = |A100047(n)|. - Michael Somos, May 24 2015 EXAMPLE G.f. = x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^12 + ... MAPLE seq(n&^4 mod 5, n=0..50); # Gary Detlefs, Mar 20 2010 MATHEMATICA Mod[#, 2]&/@CoefficientList[Series[(x+x^3)/(1+x+x^2+x^3+x^4) , {x, 0, 100}], x] (* or *) Flatten[Table[{0, 1, 1, 1, 1}, {30}]] (* Harvey P. Dale, May 15 2011 *) a[ n_] := Sign@Mod[ n, 5]; (* Michael Somos, May 24 2015 *) PROG (PARI) a(n)=!!(n%5) \\ Charles R Greathouse IV, Sep 23 2012 (PARI) {a(n) = n%5>0}; /* Michael Somos, May 24 2015 */ (Scheme) (define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; Antti Karttunen, Dec 21 2017 CROSSREFS Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively. Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486. Cf. A080891, A100047. Sequence in context: A100047 A226162 A080891 * A276398 A204549 A112713 Adjacent sequences: A011555 A011556 A011557 * A011559 A011560 A011561 KEYWORD nonn,mult,easy AUTHOR N. J. A. Sloane EXTENSIONS More terms from Antti Karttunen, Dec 21 2017 STATUS approved

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Last modified April 22 14:45 EDT 2024. Contains 371905 sequences. (Running on oeis4.)