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 A010051 Characteristic function of primes: 1 if n is prime else 0. 846
 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The following sequences all have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002 Hardy and Wright prove that the real number 0.011010100010 ... is irrational. See Nasehpour link. - Michel Marcus, Jun 21 2018 REFERENCES J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 3. V. Brun, Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare, Arch. Mat. Natur. B, 34, no. 8, 1915. Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65. LINKS N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane) Y. Motohashi, An overview of the Sieve Method and its History, arXiv:math/0505521 [math.NT], 2005-2006. Peyman Nasehpour, A Simple Criterion for Irrationality of Some Real Numbers, arXiv:1806.07560 [math.AC], 2018. J. L. Ramírez, G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014). Eric Weisstein's World of Mathematics, Prime Number Eric Weisstein's World of Mathematics, Prime Constant Eric Weisstein's World of Mathematics, Prime zeta function primezeta(s). FORMULA a(n) = floor(cos(Pi*((n-1)! + 1)/n)^2) for n >= 2. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 07 2002 Let M(n) be the n X n matrix m(i, j) = 0 if n divides ij + 1, m(i, j) = 1 otherwise; then for n > 0 a(n) = -det(M(n)). - Benoit Cloitre, Jan 17 2003 n >= 2, a(n) = floor(phi(n)/(n - 1)) = floor(A000010(n)/(n - 1)). - Benoit Cloitre, Apr 11 2003 a(n) = Sum_{d|gcd(n, A034386(n))} mu(d). [Brun] a(m*n) = a(m)*0^(n - 1) + a(n)*0^(m - 1). - Reinhard Zumkeller, Nov 25 2004 a(n) = 1 if n has no divisors other than 1 and n, and 0 otherwise. - Jon Perry, Jul 02 2005 Dirichlet generating function: Sum_{n >= 1} a(n)/n^s = primezeta(s), where primezeta is the prime zeta function. - Franklin T. Adams-Watters, Sep 11 2005 a(n) = (n-1)!^2 mod n. - Franz Vrabec, Jun 24 2006 a(n) = A047886(n, 1). - Reinhard Zumkeller, Apr 15 2008 Equals A051731 (the inverse Möbius transform) * A143519. - Gary W. Adamson, Aug 22 2008 a(n) = A051731((n + 1)! + 1, n) from Wilson's theorem: n is prime if and only if (n + 1)! is congruent to -1 mod n. - N-E. Fahssi, Jan 20 2009, Jan 29 2009 a(n) = A166260/A001477. - Mats Granvik, Oct 10 2009 a(n) = 0^A070824, where 0^0=1. - Mats Granvik, Gary W. Adamson, Feb 21 2010 It appears that a(n) = (H(n)*H(n + 1)) mod n, where H(n) = n!*Sum_{k=1..n} 1/k = A000254(n). - Gary Detlefs, Sep 12 2010 Dirichlet generating function: log( Sum_{n >= 1} 1/(A112624(n)*n^s) ). - Mats Granvik, Apr 13 2011 a(n) = A100995(n) - sqrt(A100995(n)*A193056(n)). - Mats Granvik, Jul 15 2011 a(n) * (2 - n mod 4) = A151763(n). - Reinhard Zumkeller, Oct 06 2011 (n - 1)*a(n) = ( (2n + 1)!! * Sum_{k=1..n}(1/(2*k + 1))) mod n, n > 2. - Gary Detlefs, Oct 07 2011 For n > 1, a(n) = floor(1/A001222(n)). - Enrique Pérez Herrero, Feb 23 2012 a(n) = mu(n) * Sum_{d|n} mu(d)*omega(d), where mu is A008683 and omega A001222 or A001221 indistinctly. - Enrique Pérez Herrero, Jun 06 2012 a(n) = A003418(n+1)/A003418(n) - A217863(n+1)/A217863(n) = A014963(n) - A072211(n). - Eric Desbiaux, Nov 25 2012 For n > 1, a(n) = floor(A014963(n)/n). - Eric Desbiaux, Jan 08 2013 a(n) = ((abs(n-2))! mod n) mod 2. - Timothy Hopper, May 25 2015 a(n) = abs(F(n)) - abs(F(n)-1/2) - abs(F(n)-1) + abs(f(n)-3/2), where F(n) = Sum_{m=2..n+1} (abs(1-(n)mod m) - abs(1/2-(n)mod m) + 1/2), n > 0. F(n) = 1 if n is prime, > 1 otherwise, except F(1) = 0. a(n) = 1 if F(n) = 1, 0 otherwise. - Timothy Hopper, Jun 16 2015 For n > 4, a(n) = (n-2)! mod n. - Thomas Ordowski, Jul 24 2016 From Ilya Gutkovskiy, Jul 24 2016: (Start) G.f.: A(x) = Sum_{n>=1} x^A000040(n) = B(x)*(1 - x), where B(x) is the g.f. for A000720. a(n) = floor(2/A000005(n)), for n>1. (End) a(n) = pi(n) - pi(n-1) = A000720(n) - A000720(n-1), for n>=1. - G. C. Greubel, Jan 05 2017 MAPLE A010051:= n -> if isprime(n) then 1 else 0 fi; MATHEMATICA Table[ If[ PrimeQ[n], 1, 0], {n, 105}] (* Robert G. Wilson v, Jan 15 2005 *) Table[Boole[PrimeQ[n]], {n, 105}] (* Alonso del Arte, Aug 09 2011 *) Table[PrimePi[n] - PrimePi[n-1], {n, 1, 50}] (* G. C. Greubel, Jan 05 2017 *_ PROG (MAGMA) s:=[]; for n in [1..100] do if IsPrime(n) then s:=Append(s, 1); else s:=Append(s, 0); end if; end for; s; (MAGMA) [IsPrime(n) select 1 else 0: n in [1..100]];  // Bruno Berselli, Mar 02 2011 (PARI) { for (n=1, 20000, if (isprime(n), a=1, a=0); write("b010051.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 15 2009 (PARI) a(n)=isprime(n) \\ Charles R Greathouse IV, Apr 16, 2011 (Haskell) import Data.List (unfoldr) a010051 :: Integer -> Int a010051 n = a010051_list !! (fromInteger n-1) a010051_list = unfoldr ch (1, a000040_list) where    ch (i, ps'@(p:ps)) = Just (fromEnum (i == p),                               (i + 1, if i == p then ps else ps')) -- Reinhard Zumkeller, Apr 17 2012, Sep 15 2011 CROSSREFS Cf. A051006 (constant 0.4146825... (base 10) = 0.01101010001010001010... (base 2)), A001221 (inverse Moebius transform), A143519, A156660, A156659, A156657, A059500, A053176, A059456, A072762. First differences of A000720, so A000720 gives partial sums. Column k=1 of A117278. Characteristic function of A000040. Sequence in context: A118247 A122257 A129950 * A131929 A100821 A139689 Adjacent sequences:  A010048 A010049 A010050 * A010052 A010053 A010054 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified November 21 00:36 EST 2018. Contains 317427 sequences. (Running on oeis4.)