1,1

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

The spectral components (excluding the zero frequency) of the Fourier transform of the partial sequences {a(j)} with j=1..n and n an even number, exhibit a remarkable symmetry with respect to the central frequency component at position 1 + n/4. See the Fourier spectrum of the first 2^20 terms in Links, Comments in A289777, and Conjectures in A001223 of Sep 01 2019. It also appears that the symmetry grows with n. - Andres Cicuttin, Aug 23 2020

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.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, London, 1975.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 65.

N. J. A. Sloane and Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from N. J. A. Sloane)

Andres Cicuttin, Fourier spectrum of the first 2^20 terms of the characteristic function of primes

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, Journal of Algorithms and Computation, Vol. 52, No. 1 (2020), pp. 97-104, preprint, 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).

Index entries for characteristic functions

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) = ( (2*n + 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

Decimal expansion of Sum_{k>=1} (1/10)^prime(k) = 9 * Sum_{k>=1} pi(k)/10^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

A010051:= n -> if isprime(n) then 1 else 0 fi;

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, 50}] (* G. C. Greubel, Jan 05 2017 *)

(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

(Python)

from sympy import isprime

def A010051(n): return int(isprime(n)) # Chai Wah Wu, Jan 20 2022

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: A332219 A227625 A129950 * A358751 A252233 A283991

Adjacent sequences: A010048 A010049 A010050 * A010052 A010053 A010054

nonn,easy

N. J. A. Sloane

approved