%I M0652 N0241 #1223 Jul 25 2024 09:15:28
%S 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T 97,101,103,107,109,113,127,131,137,139,149,151,157,163,167,173,179,
%U 181,191,193,197,199,211,223,227,229,233,239,241,251,257,263,269,271
%N The prime numbers.
%C See A065091 for comments, formulas etc. concerning only odd primes. For all information concerning prime powers, see A000961. For contributions concerning "almost primes" see A002808.
%C A number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p.
%C A natural number is prime if and only if it has exactly two (positive) divisors.
%C A prime has exactly one proper positive divisor, 1.
%C The paper by Kaoru Motose starts as follows: "Let q be a prime divisor of a Mersenne number 2^p-1 where p is prime. Then p is the order of 2 (mod q). Thus p is a divisor of q - 1 and q > p. This shows that there exist infinitely many prime numbers." - Pieter Moree, Oct 14 2004
%C 1 is not a prime, for if the primes included 1, then the factorization of a natural number n into a product of primes would not be unique, since n = n*1.
%C Prime(n) and pi(n) are inverse functions: A000720(a(n)) = n and a(n) is the least number m such that a(A000720(m)) = a(n). a(A000720(n)) = n if (and only if) n is prime.
%C Second sequence ever computed by electronic computer, on EDSAC, May 09 1949 (see Renwick link). - _Russ Cox_, Apr 20 2006
%C Every prime p > 3 is a linear combination of previous primes prime(n) with nonzero coefficients c(n) and |c(n)| < prime(n). - _Amarnath Murthy_, _Franklin T. Adams-Watters_ and _Joshua Zucker_, May 17 2006; clarified by _Chayim Lowen_, Jul 17 2015
%C The Greek transliteration of 'Prime Number' is 'Protos Arithmos'. - _Daniel Forgues_, May 08 2009 [Edited by _Petros Hadjicostas_, Nov 18 2019]
%C A number n is prime if and only if it is different from zero and different from a unit and each multiple of n decomposes into factors such that n divides at least one of the factors. This applies equally to the integers (where a prime has exactly four divisors (the definition of divisors is relaxed such that they can be negative)) and the positive integers (where a prime has exactly two distinct divisors). - _Peter Luschny_, Oct 09 2012
%C Motivated by his conjecture on representations of integers by alternating sums of consecutive primes, for any positive integer n, Zhi-Wei Sun conjectured that the polynomial P_n(x) = Sum_{k=0..n} a(k+1)*x^k is irreducible over the field of rational numbers with the Galois group S_n, and moreover P_n(x) is irreducible mod a(m) for some m <= n(n+1)/2. It seems that no known criterion on irreduciblity of polynomials implies this conjecture. - _Zhi-Wei Sun_, Mar 23 2013
%C Questions on a(2n) and Ramanujan primes are in A233739. - _Jonathan Sondow_, Dec 16 2013
%C From _Hieronymus Fischer_, Apr 02 2014: (Start)
%C Natural numbers such that there is exactly one base b such that the base-b alternate digital sum is 0 (see A239707).
%C Equivalently: Numbers p > 1 such that b = p-1 is the only base >= 1 for which the base-b alternate digital sum is 0.
%C Equivalently: Numbers p > 1 such that the base-b alternate digital sum is <> 0 for all bases 1 <= b < p-1. (End)
%C An integer n > 1 is a prime if and only if it is not the sum of positive integers in arithmetic progression with common difference 2. - _Jean-Christophe Hervé_, Jun 01 2014
%C Conjecture: Numbers having prime factors <= prime(n+1) are {k|k^f(n) mod primorial(n)=1}, where f(n) = lcm(prime(i)-1, i=1..n) = A058254(n) and primorial(n) = A002110(n). For example, numbers with no prime divisor <= prime(7) = 17 are {k|k^60 mod 30030=1}. - _Gary Detlefs_, Jun 07 2014
%C Cramer conjecture prime(n+1) - prime(n) < C log^2 prime(n) is equivalent to the inequality (log prime(n+1)/log prime(n))^n < e^C, as n tend to infinity, where C is an absolute constant. - _Thomas Ordowski_, Oct 06 2014
%C I conjecture that for any positive rational number r there are finitely many primes q_1,...,q_k such that r = Sum_{j=1..k} 1/(q_j-1). For example, 2 = 1/(2-1) + 1/(3-1) + 1/(5-1) + 1/(7-1) + 1/(13-1) with 2, 3, 5, 7 and 13 all prime, 1/7 = 1/(13-1) + 1/(29-1) + 1/(43-1) with 13, 29 and 43 all prime, and 5/7 = 1/(3-1) + 1/(7-1) + 1/(31-1) + 1/(71-1) with 3, 7, 31 and 71 all prime. - _Zhi-Wei Sun_, Sep 09 2015
%C I also conjecture that for any positive rational number r there are finitely many primes p_1,...,p_k such that r = Sum_{j=1..k} 1/(p_j+1). For example, 1 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(11+1) + 1/(23+1) with 2, 3, 5, 7, 11 and 23 all prime, and 10/11 = 1/(2+1) + 1/(3+1) + 1/(5+1) + 1/(7+1) + 1/(43+1) + 1/(131+1) + 1/(263+1) with 2, 3, 5, 7, 43, 131 and 263 all prime. - _Zhi-Wei Sun_, Sep 13 2015
%C Numbers k such that ((k-2)!!)^2 == +-1 (mod k). - _Thomas Ordowski_, Aug 27 2016
%C Does not satisfy Benford's law [Diaconis, 1977; Cohen-Katz, 1984; Berger-Hill, 2017]. - _N. J. A. Sloane_, Feb 07 2017
%C Prime numbers are the integer roots of 1 - sin(Pi*Gamma(s)/s)/sin(Pi/s). - _Peter Luschny_, Feb 23 2018
%C Conjecture: log log a(n+1) - log log a(n) < 1/n. - _Thomas Ordowski_, Feb 17 2023
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%D E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.
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%D M. Dietzfelbinger, Primality Testing in Polynomial Time, Springer NY 2004.
%D M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 5.
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%D W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Chap. 6.
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%D D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, Chap. 1.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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%H N. J. A. Sloane, <a href="/A000040/a000040.txt">Table of n, prime(n) for n = 1..100000</a>
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%H M. Ogihara & S. Radziszowski, <a href="http://www.cs.rit.edu/~spr/PUBL/primes.pdf">Agrawal-Kayal-Saxena Algorithm for Testing Primality in Polynomial Time</a>
%H P. Papaphilippou, <a href="http://www.philippos.info/unit_c/primeg/">Plotter of prime numbers frequency graph (flash object)</a> [From Philippos Papaphilippou (philippos(AT)safe-mail.net), Jun 02 2010]
%H J. M. Parganin, <a href="http://noe-education.org/D11102.php">Primes less than 50000</a>
%H Matthew Parker, <a href="https://oeis.org/A000040/a000040_1B.7z">The first billion primes (7-Zip compressed file)</a> [a large file]
%H Ed Pegg, Jr., <a href="http://www.mathpuzzle.com/MAA/07-Sequence%20Pictures/mathgames_12_08_03.html">Sequence Pictures</a>, Math Games column, Dec 08 2003. [<a href="/A000043/a000043_2.pdf">Cached copy, with permission (pdf only)</a>]
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%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/4.jpg">Illustration of initial terms</a>
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%H Popular Computing (Calabasas, CA), <a href="/A003309/a003309.pdf">Sieves: Problem 43</a>, Vol. 2 (No. 13, Apr 1974), pp. 6-7. [Annotated and scanned copy]
%H Primefan, <a href="http://primefan.tripod.com/500Primes1.html">The First 500 Prime Numbers</a> and <a href="http://primefan.tripod.com/PrimeLister.html">Script to Calculate Prime Numbers</a>.
%H Project Gutenberg Etext, <a href="http://www.gutenberg.org/etext/65">First 100,000 Prime Numbers</a>
%H C. D. Pruitt, <a href="http://www.mathematical.com/mathprimegen.html">Formulae for Generating All Prime Numbers</a>
%H R. Ramachandran, Frontline 19 (17) 08-2000, <a href="http://www.flonnet.com/fl1917/19171290.htm">A Prime Solution</a>
%H W. S. Renwick, <a href="http://www.cl.cam.ac.uk/Relics/elog.html">EDSAC log</a>.
%H F. Richman, <a href="http://math.fau.edu/Richman/primes.htm">Generating primes by the sieve of Eratosthenes</a>
%H Barkley Rosser, <a href="http://www.jstor.org/stable/2371291">Explicit Bounds for Some Functions of Prime Numbers</a>, American Journal of Mathematics 63 (1941) 211-232.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807"> Approximate formulas for some functions of prime numbers</a>, Illinois J. Math. Volume 6, Issue 1 (1962), 64-94.
%H S. M. Ruiz and J. Sondow, <a href="http://arXiv.org/abs/math.NT/0210312">Formulas for pi(n) and the n-th prime</a>, arXiv:math/0210312 [math.NT], 2002-2014.
%H S. O. S. Math, <a href="http://www.sosmath.com/tables/prime/prime.html">First 1000 Prime Numbers</a>
%H A. Schulman, <a href="http://www.sonic.net/~undoc/java/PrimeCalc.html">Prime Number Calculator</a>
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 5.
%H M. Slone, PlanetMath.Org, <a href="http://planetmath.org/encyclopedia/FirstThousandPositivePrimeNumbers.html">First thousand positive prime numbers</a>
%H A. Stiglic, <a href="http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html">The PRIMES is in P little FAQ</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2013.02.003">On functions taking only prime values</a>, J. Number Theory, 133 (2013), no. 8, 2794-2812.
%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/UnitFraction.pdf">A conjecture on unit fractions involving primes</a>, Preprint 2015.
%H J. Teitelbaum, <a href="http://www.ams.org/bull/2002-39-03/S0273-0979-02-00947-3/S0273-0979-02-00947-3.pdf">Review of "Prime numbers:A computational perspective" by R. Crandall & C. Pomerance</a>
%H J. Thonnard, <a href="http://www.proftnj.com/calcprem.htm">Les nombres premiers(Primality check; Closest next prime; Factorizer)</a>
%H J. Tramu, <a href="http://www.echolalie.org/gbgraph/gb/index.htm">Movie of primes scrolling</a>
%H A. Turpel, <a href="http://www2.vo.lu/homepages/armand/index.html">Aesthetics of the Prime Sequence</a> [broken link ?]
%H S. Wagon, <a href="http://www.americanscientist.org/bookshelf/pub/prime-time">Prime Time: Review of "Prime Numbers:A Computational Perspective" by R. Crandall & C. Pomerance</a>
%H M. R. Watkins, <a href="http://www.maths.ex.ac.uk/~mwatkins/zeta/unusual.htm">unusual and physical methods for finding prime numbers</a>
%H S. Wedeniwski, <a href="http://w210.Ub.uni-tuebingen.de/dbt/volltexte/2001/420/pdf/dissertation.pdf">Primality Tests on Commutator Curves</a>
%H E. Wegrzynowski, <a href="https://web.archive.org/web/20160413222527/http://www.lifl.fr/~wegrzyno/FormulPrem/FormulesPremiers23.html">Les formules simples qui donnent des nombres premiers en grande quantité</a> (in French).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-Generating Polynomial</a>, <a href="http://mathworld.wolfram.com/PrimeNumber.html">Prime Number</a>, and <a href="http://mathworld.wolfram.com/PrimeSpiral.html">Prime Spiral</a>.
%H Wikipedia, <a href="https://www.wikipedia.org/wiki/Prime_number">Prime number</a> and <a href="https://www.wikipedia.org/wiki/Prime_number_theorem">Prime number theorem</a>.
%H G. Xiao, Primes server, <a href="http://wims.unice.fr/~wims/en_tool~number~primes.html">Sequential Batches Primes Listing (up to orders not exceeding 10^308)</a>
%H G. Xiao, Numerical Calculator, <a href="http://wims.unice.fr/wims/en_tool~number~calcnum.en.html">To display p(n) for n up to 41561, operate on "prime(n)"</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%H <a href="/A000040/a000040_1.html">Additional (less important) items -- comments, formulas, references, links, programs, etc. -- related to the prime numbers, A000040. [HTML version]</a> - <a href="/A000040/a000040_2.txt">[Plain text (TXT) version]</a>.
%F The prime number theorem is the statement that a(n) ~ n * log n as n -> infinity (Hardy and Wright, page 10).
%F For n >= 2, n*(log n + log log n - 3/2) < a(n); for n >= 20, a(n) < n*(log n + log log n - 1/2). [Rosser and Schoenfeld]
%F For all n, a(n) > n log n. [Rosser]
%F n log(n) + n (log log n - 1) < a(n) < n log n + n log log n for n >= 6. [Dusart, quoted in the Wikipedia article]
%F a(n) = n log n + n log log n + (n/log n)*(log log n - log n - 2) + O( n (log log n)^2/ (log n)^2). [Cipolla, see also Cesàro or the "Prime number theorem" Wikipedia article for more terms in the expansion]
%F a(n) = 2 + Sum_{k = 2..floor(2n*log(n)+2)} (1-floor(pi(k)/n)), for n > 1, where the formula for pi(k) is given in A000720 (Ruiz and Sondow 2002). - _Jonathan Sondow_, Mar 06 2004
%F I conjecture that Sum_{i>=1} (1/(prime(i)*log(prime(i)))) = Pi/2 = 1.570796327...; Sum_{i=1..100000} (1/(prime(i)*log(prime(i)))) = 1.565585514... It converges very slowly. - _Miklos Kristof_, Feb 12 2007
%F The last conjecture has been discussed by the math.research newsgroup recently. The sum, which is greater than Pi/2, is shown in sequence A137245. - _T. D. Noe_, Jan 13 2009
%F A000005(a(n)) = 2; A002033(a(n+1)) = 1. - _Juri-Stepan Gerasimov_, Oct 17 2009
%F A001222(a(n)) = 1. - _Juri-Stepan Gerasimov_, Nov 10 2009
%F From _Gary Detlefs_, Sep 10 2010: (Start)
%F Conjecture:
%F a(n) = {n| n! mod n^2 = n(n-1)}, n <> 4.
%F a(n) = {n| n!*h(n) mod n = n-1}, n <> 4, where h(n) = Sum_{k=1..n} 1/k. (End)
%F For n = 1..15, a(n) = p + abs(p-3/2) + 1/2, where p = m + int((m-3)/2), and m = n + int((n-2)/8) + int((n-4)/8). - _Timothy Hopper_, Oct 23 2010
%F a(2n) <= A104272(n) - 2 for n > 1, and a(2n) ~ A104272(n) as n -> infinity. - _Jonathan Sondow_, Dec 16 2013
%F Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(n-1) mod n = 1}. - _Gary Detlefs_, May 25 2014
%F Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(3*n) mod 3*n = 8}. - _Gary Detlefs_, May 28 2014
%F a(n) = 1 + Sum_{m=1..L(n)} (abs(n-Pi(m))-abs(n-Pi(m)-1/2)+1/2), where Pi(m) = A000720(m) and L(n) >= a(n)-1. L(n) can be any function of n which satisfies the inequality. For instance, L(n) can be ceiling((n+1)*log((n+1)*log(n+1))) since it satisfies this inequality. - _Timothy Hopper_, May 30 2015, Jun 16 2015
%F Satisfies a(n) = 2*n + Sum_{k=1..(a(n)-1)} cot(k*Pi/a(n))*sin(2*k*n^a(n)*Pi/a(n)). - _Ilya Gutkovskiy_, Jun 29 2016
%F Sum_{n>=1} 1/a(n)^s = P(s), where P(s) is the prime zeta function. - _Eric W. Weisstein_, Nov 08 2016
%F a(n) = floor(1 - log(-1/2 + Sum_{ d | A002110(n-1) } mu(d)/(2^d-1))/log(2)) where mu(d) = A008683(d). Golomb gave a proof in 1974: Give each positive integer a probability of W(n) = 1/2^n, then the probability M(d) of the integer multiple of number d equals 1/(2^d-1). Suppose Q = a(1)*a(2)*...*a(n-1) = A002110(n-1), then the probability of random integers that are mutually prime with Q is Sum_{ d | Q } mu(d)*M(d) = Sum_{ d | Q } mu(d)/(2^d-1) = Sum_{ gcd(m, Q) = 1 } W(m) = 1/2 + 1/2^a(n) + 1/2^a(n+1) + 1/2^a(n+2) + ... So ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) = 1 + x(n), which means that a(n) is the only integer so that 1 < ((Sum_{ d | Q } mu(d)/(2^d-1)) - 1/2)*2^a(n) < 2. - _Jinyuan Wang_, Apr 08 2019
%p A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
%p # For illustration purposes only:
%p isPrime := s -> is(1 = sin(Pi*GAMMA(s)/s)/sin(Pi/s)):
%p select(isPrime, [$2..100]); # _Peter Luschny_, Feb 23 2018
%t Prime[Range[60]]
%o (Magma) [n : n in [2..500] | IsPrime(n)];
%o (Magma) a := func< n | NthPrime(n) >;
%o (PARI) {a(n) = if( n<1, 0, prime(n))};
%o (PARI) /* The following functions provide asymptotic approximations, one based on the asymptotic formula cited above (slight overestimate for n > 10^8), the other one based on pi(x) ~ li(x) = Ei(log(x)) (slight underestimate): */
%o prime1(n)=n*(log(n)+log(log(n))-1+(log(log(n))-2)/log(n)-((log(log(n))-6)*log(log(n))+11)/log(n)^2/2)
%o prime2(n)=solve(X=n*log(n)/2,2*n*log(n),real(eint1(-log(X)))+n)
%o \\ _M. F. Hasler_, Oct 21 2013
%o (PARI) forprime(p=2, 10^3, print1(p, ", ")) \\ _Felix Fröhlich_, Jun 30 2014
%o (PARI) primes(10^5) \\ _Altug Alkan_, Mar 26 2018
%o (Sage) a = sloane.A000040
%o a.list(58) # _Jaap Spies_, 2007
%o (Sage) prime_range(1, 300) # _Zerinvary Lajos_, May 27 2009
%o (Maxima) A000040(n) := block(
%o if n = 1 then return(2),
%o return( next_prime(A000040(n-1)))
%o )$ /* recursive, to be replaced if possible - _R. J. Mathar_, Feb 27 2012 */
%o (Haskell) See also Haskell Wiki Link
%o import Data.List (genericIndex)
%o a000040 n = genericIndex a000040_list (n - 1)
%o a000040_list = base ++ larger where
%o base = [2,3,5,7,11,13,17]
%o larger = p : filter prime more
%o prime n = all ((> 0) . mod n) $ takeWhile (\x -> x*x <= n) larger
%o _ : p : more = roll $ makeWheels base
%o roll (Wheel n rs) = [n * k + r | k <- [0..], r <- rs]
%o makeWheels = foldl nextSize (Wheel 1 [1])
%o nextSize (Wheel size bs) p = Wheel (size * p)
%o [r | k <- [0..p-1], b <- bs, let r = size*k+b, mod r p > 0]
%o data Wheel = Wheel Integer [Integer]
%o -- _Reinhard Zumkeller_, Apr 07 2014
%o (GAP)
%o A000040:=Filtered([1..10^5],IsPrime); # _Muniru A Asiru_, Sep 04 2017
%o (Python)
%o from sympy import primerange
%o print(list(primerange(2, 272))) # _Michael S. Branicky_, Apr 30 2022
%Y For is_prime and next_prime, see A010051 and A151800.
%Y Cf. A000720 ("pi"), A001223 (differences between primes), A002476, A002808, A003627, A006879, A006880, A008578, A233588.
%Y Cf. primes in lexicographic order: A210757, A210758, A210759, A210760, A210761.
%Y Cf. A003558, A179480 (relating to the Quasi-order theorem of Hilton and Pedersen).
%Y Boustrophedon transforms: A000747, A000732, A230953.
%Y a(2n) = A104272(n) - A233739(n).
%K core,nonn,nice,easy
%O 1,1
%A _N. J. A. Sloane_
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