A008578 -id:A008578 - cp's OEIS frontend

A000040 The prime numbers.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271

Offset: 1

N. J. A. Sloane

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.
A number p is prime if (and only if) it is greater than 1 and has no positive divisors except 1 and p.
A natural number is prime if and only if it has exactly two (positive) divisors.
A prime has exactly one proper positive divisor, 1.
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
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.
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.
Second sequence ever computed by electronic computer, on EDSAC, May 09 1949 (see Renwick link). - Russ Cox, Apr 20 2006
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
The Greek transliteration of 'Prime Number' is 'Protos Arithmos'. - Daniel Forgues, May 08 2009 [Edited by Petros Hadjicostas, Nov 18 2019]
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
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 irreducibility of polynomials implies this conjecture. - Zhi-Wei Sun, Mar 23 2013
Questions on a(2n) and Ramanujan primes are in A233739. - Jonathan Sondow, Dec 16 2013
From Hieronymus Fischer, Apr 02 2014: (Start)
Natural numbers such that there is exactly one base b such that the base-b alternate digital sum is 0 (see A239707).
Equivalently: Numbers p > 1 such that b = p-1 is the only base >= 1 for which the base-b alternate digital sum is 0.
Equivalently: Numbers p > 1 such that the base-b alternate digital sum is <> 0 for all bases 1 <= b < p-1. (End)
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
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
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
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
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
Numbers k such that ((k-2)!!)^2 == +-1 (mod k). - Thomas Ordowski, Aug 27 2016
Does not satisfy Benford's law [Diaconis, 1977; Cohen-Katz, 1984; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 07 2017
Prime numbers are the integer roots of 1 - sin(Pi*Gamma(s)/s)/sin(Pi/s). - Peter Luschny, Feb 23 2018
Conjecture: log log a(n+1) - log log a(n) < 1/n. - Thomas Ordowski, Feb 17 2023

			From _David A. Corneth_, Oct 22 2024: (Start)
7 is a prime number as it has exactly two divisors, 1 and 7.
8 is not a prime number as it does not have exactly two divisors (it has 1, 2, 4 and 8 as divisors though it is sufficient to find one other divisor than 1 and 8)
55 is not a prime number as it does not have exactly two divisors. One other divisor than 1 and 55 is 5.
59 is a prime number as it has exactly two divisors; 1 and 59. (End)

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M. Agrawal, A Short History of "PRIMES is in P"
P. Alfeld, Notes and Literature on Prime Numbers
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Anonymous, Prime Number Master Index (for primes up to 2*10^7)
Anonymous, prime number
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Christian Axler, New estimates for the n-th prime number, arXiv:1706.03651 [math.NT], 2017.
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Index entries for "core" sequences
Index entries for sequences related to Benford's law
Additional (less important) items -- comments, formulas, references, links, programs, etc. -- related to the prime numbers, A000040. [HTML version] - [Plain text (TXT) version].

For is_prime and next_prime, see A010051 and A151800.
Cf. A000720 ("pi"), A001223 (differences between primes), A002476, A002808, A003627, A006879, A006880, A008578, A080339, A233588.
Cf. primes in lexicographic order: A210757, A210758, A210759, A210760, A210761.
Cf. A003558, A179480 (relating to the Quasi-order theorem of Hilton and Pedersen).
Boustrophedon transforms: A000747, A000732, A230953.
a(2n) = A104272(n) - A233739(n).
Related sequences:
Primes (p) and composites (c): A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A000040:=Filtered([1..10^5],IsPrime); # Muniru A Asiru, Sep 04 2017

-- See also Haskell Wiki Link.
import Data.List (genericIndex)
a000040 n = genericIndex a000040_list (n - 1)
a000040_list = base ++ larger where
base = [2,3,5,7,11,13,17]
larger = p : filter prime more
prime n = all ((> 0) . mod n) $ takeWhile (\x -> x*x <= n) larger
_ : p : more = roll $ makeWheels base
roll (Wheel n rs) = [n * k + r | k <- [0..], r <- rs]
makeWheels = foldl nextSize (Wheel 1 [1])
nextSize (Wheel size bs) p = Wheel (size * p)
[r | k <- [0..p-1], b <- bs, let r = size*k+b, mod r p > 0]
data Wheel = Wheel Integer [Integer]
-- Reinhard Zumkeller, Apr 07 2014

Magma
```
[n : n in [2..500] | IsPrime(n)];
```
Magma
```
a := func< n | NthPrime(n) >;
```

A000040 := n->ithprime(n); [ seq(ithprime(i),i=1..100) ];
# For illustration purposes only:
isPrime := s -> is(1 = sin(Pi*GAMMA(s)/s)/sin(Pi/s)):
select(isPrime, [$2..100]); # Peter Luschny, Feb 23 2018

Mathematica
```
Prime[Range[60]]
```

A000040(n) := block(
if n = 1 then return(2),
return( next_prime(A000040(n-1)))
)$ /* recursive, to be replaced if possible - R. J. Mathar, Feb 27 2012 */

PARI
```
{a(n) = if( n<1, 0, prime(n))};
```

/* 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): */
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)
prime2(n)=solve(X=n*log(n)/2,2*n*log(n),real(eint1(-log(X)))+n)
\\ M. F. Hasler, Oct 21 2013

forprime(p=2, 10^3, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014

primes(10^5) \\ Altug Alkan, Mar 26 2018

from sympy import primerange
print(list(primerange(2, 272))) # Michael S. Branicky, Apr 30 2022

a = sloane.A000040
a.list(58)  # Jaap Spies, 2007

prime_range(1, 300)  # Zerinvary Lajos, May 27 2009

The prime number theorem is the statement that a(n) ~ n * log n as n -> infinity (Hardy and Wright, page 10).
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]
For all n, a(n) > n log n. [Rosser]
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]
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]
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
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
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
A000005(a(n)) = 2; A002033(a(n+1)) = 1. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) = 1. - Juri-Stepan Gerasimov, Nov 10 2009
From Gary Detlefs, Sep 10 2010: (Start)
Conjecture:
a(n) = {n| n! mod n^2 = n(n-1)}, n <> 4.
a(n) = {n| n!*h(n) mod n = n-1}, n <> 4, where h(n) = Sum_{k=1..n} 1/k. (End)
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
a(2n) <= A104272(n) - 2 for n > 1, and a(2n) ~ A104272(n) as n -> infinity. - Jonathan Sondow, Dec 16 2013
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
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
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
Sum_{n>=1} 1/a(n)^s = P(s), where P(s) is the prime zeta function. - Eric W. Weisstein, Nov 08 2016
a(n) = floor(1 - log(-1/2 + Sum_{ d | A002110(n-1) } mu(d)/(2^d-1))/log(2)) where mu(d) = A008683(d) [Ghandi, 1971] (see Ribenboim). 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
Conjecture: n * (log(n)+log(log(n))-1+((log(log(n))-A)/log(n))) is asymptotic to a(n) if and only if A=2. - Alain Rocchelli, Feb 12 2025
From Stefano Spezia, Apr 13 2025: (Start)
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/Sum_{j=1..m} A080339(j))^(1/n)) [Willans, 1964].
a(n) = 1 + Sum_{m=1..2^n} floor(floor(n/(1 + A000720(m)))^(1/n)) [Willans, 1964]. (End)

A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 3, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 3, 37, 19, 13, 5, 41, 7, 43, 11, 5, 23, 47, 3, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 3, 73, 37, 5, 19, 11, 13, 79, 5, 3, 41, 83, 7, 17, 43

Offset: 1

N. J. A. Sloane

The initial term a(1)=1 is purely conventional: The unit 1 is not a prime number, although it has been considered so in the past. 1 is the empty product of prime numbers, thus 1 has no largest prime factor. - Daniel Forgues, Jul 05 2011
Greatest noncomposite number dividing n, (cf. A008578). - Omar E. Pol, Aug 31 2013
Conjecture: Let a, b be nonzero integers and f(n) denote the maximum prime factor of a*n + b if a*n + b <> 0 and f(n)=0 if a*n + b=0 for any integer n. Then the set {n, f(n), f(f(n)), ...} is finite of bounded size. - M. Farrokhi D. G., Jan 10 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section IV.1.
H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, Amer. Math. Soc., 1996, p. 210.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294. MR 0447086 (56 #5401).
Greg Back and Mihai Caragiu, The greatest prime factor and recurrent sequences, Fib. Q., 48 (2010), 358-362.
Thomas Bloom, Problem 683 and Problem 928, Erdős Problems.
Andries E. Brouwer, Two number theoretic sums, Stichting Mathematisch Centrum. Zuivere Wiskunde, Report ZW 19/74 (1974): 3 pages. [Cached copy, included with the permission of the author]
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Mats Granvik, Mathematica program to check the conjectured formula, Feb 28 2021.
Nathan McNew, The Most Frequent Values of the Largest Prime Divisor Function, Exper. Math., 2017, Vol. 26, No. 2, 210-224.
OEIS Wiki, Greatest prime factor of n
Herman P. Robinson, Letter to N. J. A. Sloane, Oct 1981
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
Terence Tao, Erdős problem database, see nos. 683 and 928.
Eric Weisstein's World of Mathematics, Greatest Prime Factor
Index entries for "core" sequences

Cf. A000040, A020639 (smallest prime divisor), A034684, A028233, A034699, A053585.
See also A032742, A052126, A070087, A070089, A061395, A175723.
Cf. A046670 (partial sums), A104350 (partial products).
See A385503 for "popular" primes.

[ #f eq 0 select 1 else f[ #f][1] where f is Factorization(n): n in [1..86] ]; // Klaus Brockhaus, Oct 23 2008

with(numtheory,divisors); A006530 := proc(n) local i,t1,t2,t3,t4,t5; t1 := divisors(n); t2 := convert(t1,list); t3 := sort(t2); t4 := nops(t3); t5 := 1; for i from 1 to t4 do if isprime(t3[t4+1-i]) then return t3[t4+1-i]; fi; od; 1; end;
# alternative
A006530 := n->max(1,op(numtheory[factorset](n))); # Peter Luschny, Nov 02 2010

Table[ FactorInteger[n][[ -1, 1]], {n, 100}] (* Ray Chandler, Nov 12 2005 and modified by Robert G. Wilson v, Jul 16 2014 *)

A006530(n)=if(n>1,vecmax(factor(n)[,1]),1) \\ Edited to cover n=1. - M. F. Hasler, Jul 30 2015

from sympy import factorint
def a(n): return 1 if n == 1 else max(factorint(n))
print([a(n) for n in range(1, 87)]) # Michael S. Branicky, Aug 08 2022

def A006530(n): return list(factor(n))[-1][0] if n > 1 else 1
print([A006530(n) for n in range(1, 87)])  # Peter Luschny, Jan 07 2024

;; The following uses macro definec for the memoization (caching) of the results. A naive implementation of A020639 can be found under that entry. It could be also defined with definec to make it faster on the later calls. See http://oeis.org/wiki/Memoization#Scheme
(definec (A006530 n) (let ((spf (A020639 n))) (if (= spf n) spf (A006530 (/ n spf)))))
;; Antti Karttunen, Mar 12 2017

a(n) = A027748(n, A001221(n)) = A027746(n, A001222(n)); a(n)^A071178(n) = A053585(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A000040(A061395(n)). - M. F. Hasler, Jan 16 2015
a(n) = n + 1 - Sum_{k=1..n} (floor((k!^n)/n) - floor(((k!^n)-1)/n)). - Anthony Browne, May 11 2016
n/a(n) = A052126(n). - R. J. Mathar, Oct 03 2016
If A020639(n) = n [when n is 1 or a prime] then a(n) = n, otherwise a(n) = a(A032742(n)). - Antti Karttunen, Mar 12 2017
a(n) has average order Pi^2*n/(12 log n) [Brouwer]. See also A046670. - N. J. A. Sloane, Jun 26 2017

Edited by M. F. Hasler, Jan 16 2015

A002808 The composite numbers: numbers n of the form x*y for x > 1 and y > 1.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

The natural numbers 1,2,... are divided into three sets: 1 (the unit), the primes (A000040) and the composite numbers (A002808).
The number of composite numbers <= n (A065855) = n - pi(n) (A000720) - 1.
n is composite iff sigma(n) + phi(n) > 2n. This is a nice result of the well known theorem: For all positive integers n, n = Sum_{d|n} phi(d). For the proof see my contribution to puzzle 76 of Carlos Rivera's Primepuzzles. - Farideh Firoozbakht, Jan 27 2005, Jan 18 2015
The composite numbers have the semiprimes A001358 as primitive elements.
A211110(a(n)) > 1. - Reinhard Zumkeller, Apr 02 2012
A060448(a(n)) > 1. - Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) > 0. - Reinhard Zumkeller, Dec 14 2012
Composite numbers n which are the product of r=A001222(n) prime numbers are sometimes called r-almost primes. Sequences listing r-almost primes are: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
a(n) = A056608(n) * A160180(n). - Reinhard Zumkeller, Mar 29 2014
Degrees for which there are irreducible polynomials which are reducible mod p for all primes p, see Brandl. - Charles R Greathouse IV, Sep 04 2014
An integer is composite if and only if it is the sum of strictly positive integers in arithmetic progression with common difference 2: 4 = 1 + 3, 6 = 2 + 4, 8 = 3 + 5, 9 = 1 + 3 + 5, etc. - Jean-Christophe Hervé, Oct 02 2014
This statement holds since k+(k+2)+...+k+2(n-1) = n*(n+k-1) = a*b with arbitrary a,b (taking n=a and k=b-a+1 if b>=a). - M. F. Hasler, Oct 04 2014
For n > 4, these are numbers n such that n!/n^2 = (n-1)!/n is an integer (see A056653). - Derek Orr, Apr 16 2015
Let f(x) = Sum_{i=1..x} Sum_{j=2..i-1} cos((2*Pi*x*j)/i). It is known that the zeros of f(x) are the prime numbers. So these are the numbers n such that f(n) > 0. - Michel Lagneau, Oct 13 2015
Numbers n that can be written as solutions of the Diophantine equation n = (x+2)(y+2) where {x,y} in N^2, pairs of natural numbers including zero (cf. Mathematica code and Davis). - Ron R Spencer and Bradley Klee, Aug 15 2016
Numbers n with a partition (containing at least two summands) so that its summands also multiply to n. If n is prime, there is no way to find those two (or more) summands. If n is composite, simply take a factor or several, write those divisors and fill with enough 1's so that they add up to n. For example: 4 = 2*2 = 2+2, 6 = 1*2*3 = 1+2+3, 8 = 1*1*2*4 = 1+1+2+4, 9 = 1*1*1*3*3 = 1+1+1+3+3. - Juhani Heino, Aug 02 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 127.
Martin Davis, "Algorithms, Equations, and Logic", pp. 4-15 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
R. K. Guy, Unsolved Problems Number Theory, Section A.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 66.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 51.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..17737 [composites up to 20000]
Rolf Brandl, Integer polynomials that are reducible modulo all primes, Amer. Math. Monthly 93 (1986), pp. 286-288.
C. K. Caldwell, Composite Numbers
Felix Huber, Illustration for a(1)-a(12)
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
Carlos Rivera, Puzzle 76, z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94
Eric Weisstein's World of Mathematics, Composite Number
Index entries for "core" sequences
Index entries for sequences from "Goedel, Escher, Bach"

Complement of A008578. - Omar E. Pol, Dec 16 2016
Cf. A000040, A018252, A065090, A136527.
Cf. A073783 (first differences), A073445 (second differences).
Boustrophedon transforms: A230954, A230955.
Cf. A163870 (nontrivial divisors).
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a002808 n = a002808_list !! (n-1)
a002808_list = filter ((== 1) . a066247) [2..]
-- Reinhard Zumkeller, Feb 04 2012

[n: n in [2..250] | not IsPrime(n)]; // G. C. Greubel, Feb 24 2024

t := []: for n from 2 to 20000 do if isprime(n) then else t := [op(t),n]; fi; od: t; remove(isprime,[$3..89]); # Zerinvary Lajos, Mar 19 2007
A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do ; end if; end proc; # R. J. Mathar, Oct 27 2009

Select[Range[2,100], !PrimeQ[#]&] (* Zak Seidov, Mar 05 2011 *)
With[{nn=100},Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, May 01 2012 *)
Select[Range[100], CompositeQ] (* Jean-François Alcover, Nov 07 2021 *)

A002808(n)=for(k=0,primepi(n),isprime(n++)&&k--);n \\ For illustration only: see below. - M. F. Hasler, Oct 31 2008

A002808(n)= my(k=-1); while(-n + n += -k + k=primepi(n),); n \\ For n=10^4 resp. 3*10^4, this is about 100 resp. 500 times faster than the former; M. F. Hasler, Nov 11 2009

forcomposite(n=1, 1e2, print1(n, ", ")) \\ Felix Fröhlich, Aug 03 2014

for(n=1, 1e3, if(bigomega(n) > 1, print1(n, ", "))) \\ Altug Alkan, Oct 14 2015

from sympy import primepi
def A002808(n):
    m, k = n, primepi(n) + 1 + n
    while m != k:
        m, k = k, primepi(k) + 1 + n
    return m # Chai Wah Wu, Jul 15 2015, updated Apr 14 2016

from sympy import isprime
def ok(n): return n > 1 and not isprime(n)
print([k for k in range(89) if ok(k)]) # Michael S. Branicky, Nov 07 2021

next_A002808=lambda n: next(n for n in range(n,n*5)if not isprime(n)) # next composite >= n > 0; next_A002808(n)==n <=> iscomposite(n). - M. F. Hasler, Mar 28 2025
is_A002808=lambda n:not isprime(n) and n>1 # where isprime(n) can be replaced with: all(n%d for d in range(2, int(n**.5)+1))
# generators of composite numbers:
A002808_upto=lambda stop=1<<59: filter(is_A002808, range(2,stop))
A002808_seq=lambda:(q:=2)and(n for p in primes if (o:=q)<(q:=p) for n in range(o+1,p)) # with, e.g.: primes=filter(isprime,range(2,1<<59)) # M. F. Hasler, Mar 28 2025

[n for n in (2..250) if not is_prime(n)] # G. C. Greubel, Feb 24 2024

a(n) = pi(a(n)) + 1 + n, where pi is the prime counting function.
a(n) = A136527(n, n).
A000005(a(n)) > 2. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) > 1. - Juri-Stepan Gerasimov, Oct 30 2009
A000203(a(n)) < A007955(a(n)). - Juri-Stepan Gerasimov, Mar 17 2011
A066247(a(n)) = 1. - Reinhard Zumkeller, Feb 05 2012
Sum_{n>=1} 1/a(n)^s = Zeta(s)-1-P(s), where P is prime zeta. - Enrique Pérez Herrero, Aug 08 2012
n + n/log n + n/log^2 n < a(n) < n + n/log n + 3n/log^2 n for n >= 4, see Panaitopol. Bojarincev gives an asymptotic version. - Charles R Greathouse IV, Oct 23 2012
a(n) > n + A000720(n) + 1. - François Huppé, Jan 08 2025

Deleted an incomplete and broken link. - N. J. A. Sloane, Dec 16 2010

A064989 Multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 17, 3, 10, 7, 19, 2, 9, 11, 8, 5, 23, 6, 29, 1, 14, 13, 15, 4, 31, 17, 22, 3, 37, 10, 41, 7, 12, 19, 43, 2, 25, 9, 26, 11, 47, 8, 21, 5, 34, 23, 53, 6, 59, 29, 20, 1, 33, 14, 61, 13, 38, 15, 67, 4, 71, 31, 18, 17, 35, 22, 73, 3, 16

Offset: 1

Vladeta Jovovic, Oct 30 2001

From Antti Karttunen, May 12 2014: (Start)
a(A003961(n)) = n for all n. [This is a left inverse function for the injection A003961.]
Bisections are A064216 (the terms at odd indices) and A064989 itself (the terms at even indices), i.e., a(2n) = a(n) for all n.
(End)
From Antti Karttunen, Dec 18-21 2014: (Start)
When n represents an unordered integer partition via the indices of primes present in its prime factorization (for n >= 2, n corresponds to the partition given as the n-th row of A112798) this operation subtracts one from each part. If n is of the form 2^k (a partition having just k 1's as its parts) the result is an empty partition (which is encoded by 1, having an "empty" factorization).
For all odd numbers n >= 3, a(n) tells which number is located immediately above n in square array A246278. Cf. also A246277.
(End)
Alternatively, if numbers are represented as the multiset of indices of prime factors with multiplicity, this operation subtracts 1 from each element and discards the 0's. - M. F. Hasler, Dec 29 2014

			a(20) = a(2^2*5) = a(2^2)*a(5) = prevprime(5) = 3.

Harry J. Smith (first 1000 terms) & Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A064216 (odd bisection), A003961 (inverse), A151799.
Cf. A000040, A008578, A027746, A032742, A049084, A052126, A055396, A061395, A112798, A249817, A249818.
Other sequences whose definition involve or are some other way related with this sequence: A105560, A108951, A118306, A122111, A156552, A163511, A200746, A241909, A243070, A243071, A243072, A243073, A244319, A245605, A245607, A246165, A246266, A246268, A246277, A246278, A246361, A246362, A246371, A246372, A246373, A246374, A246376, A246380, A246675, A246682, A249745, A250470.
Similar prime-shifts towards smaller numbers: A252461, A252462, A252463.

a064989 1 = 1
a064989 n = product $ map (a008578 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(MIT/GNU Scheme, with Aubrey Jaffer's SLIB Scheme library)
(require 'factor)
(define (A064989 n) (if (= 1 n) n (apply * (map (lambda (k) (if (zero? k) 1 (A000040 k))) (map -1+ (map A049084 (factor n)))))))
;; Antti Karttunen, May 12 2014
(definec (A064989 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A064989 (A032742 n))))) ;; One based on given recurrence and utilizing memoizing definec-macro.
(definec (A064989 n) (cond ((= 1 n) n) ((even? n) (A064989 (/ n 2))) (else (A163511 (/ (- (A243071 n) 1) 2))))) ;; Corresponds to one of the alternative formulas, but is very unpractical way to compute this sequence. - Antti Karttunen, Dec 18 2014

q:= proc(p) prevprime(p) end proc: q(2):= 1:
[seq(mul(q(f[1])^f[2], f = ifactors(n)[2]), n = 1 .. 1000)]; # Robert Israel, Dec 21 2014

Table[Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n, {n, 81}] (* Michael De Vlieger, Jan 04 2016 *)

{ for (n=1, 1000, f=factor(n)~; a=1; j=1; if (n>1 && f[1, 1]==2, j=2); for (i=j, length(f), a*=precprime(f[1, i] - 1)^f[2, i]); write("b064989.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 02 2009

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((p=f[i,1]) % 2, f[i,1] = precprime(p-1), f[i,1] = 1);); factorback(f);} \\ Michel Marcus, Dec 18 2014

A064989(n)=factorback(Mat(apply(t->[max(precprime(t[1]-1),1),t[2]],Vec(factor(n)~))~)) \\ M. F. Hasler, Dec 29 2014

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

from math import prod
from sympy import prevprime, factorint
def A064989(n): return prod(prevprime(p)**e for p, e in  factorint(n>>(~n&n-1).bit_length()).items()) # Chai Wah Wu, Jan 05 2023

From Antti Karttunen, Dec 18 2014: (Start)
If n = product A000040(k)^e(k) then a(n) = product A008578(k)^e(k) [where A000040(n) gives the n-th prime, and A008578(n) gives 1 for 1 and otherwise the (n-1)-th prime].
a(1) = 1; for n > 1, a(n) = A008578(A055396(n)) * a(A032742(n)). [Above formula represented as a recurrence. Cf. A252461.]
a(1) = 1; for n > 1, a(n) = A008578(A061395(n)) * a(A052126(n)). [Compare to the formula of A252462.]
This prime-shift operation is used in the definitions of many other sequences, thus it can be expressed in many alternative ways:
a(n) = A200746(n) / n.
a(n) = A242424(n) / A105560(n).
a(n) = A122111(A122111(n)/A105560(n)) = A122111(A052126(A122111(n))). [In A112798-partition context: conjugate, remove the largest part (the largest prime factor), and conjugate again.]
a(1) = 1; for n > 1, a(2n) = a(n), a(2n+1) = A163511((A243071(2n+1)-1) / 2).
a(n) = A249818(A250470(A249817(n))). [A250470 is an analogous operation for "going one step up" in the square array A083221 (A083140).]
(End)
Product_{k=1..n} a(k) = n! / A307035(n). - Vaclav Kotesovec, Mar 21 2019
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.220703928... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - Amiram Eldar, Nov 18 2022

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A008578 gives the noncomposite numbers, the prime numbers at the beginning of the 20th century which included 1.
a(2n) = 0, 0, 2, 4, 8, 12, 14, 20, 24, 24, ... always even?
a(2n+3) =  1, 3, 5, 9, 11, 17, 23, 23, 29, ... always odd?
First differences: 0, 0, 1, 1, 1, 1, 1, 3, 1, 3, -1, 3, 3, 3, 3, 1, -1, 1, 5, 7, ... .

Conjecture: this is a permutation of the natural numbers.

Generalized conjecture: Let T(n, k) = b(k) * prime(n - k + A061395(b(k))) for 1 < k <= n, and T(n, 1) = A008578(n), where b(n), n > 0, is a permutation of the natural numbers with b(1) = 1, then T(n, k), read by rows, is a permutation of the natural numbers.

As noted in A369797, a(n) is either 1 or a prime. But not mentioned is that all primes appear once, with two exceptions. The second prime, 3, never appears and the third prime, 5, appears twice, at 4 and 9.
A denominator of 1 in A369797 appears about 81% of the time.
The graph of this sequence has three broken lines, one at a(n) = 1, the second at ~3n/2 and the third at ~3n.

cp's OEIS Frontend

A206798 Beginning with the natural numbers, swap the k-th composite number (A002808) and k-th noncomposite number (A008578), for k = 1,2,3,...

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A245240 Coefficients of the series reversion of the series Sum(x^k for k in A008578).

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A275771 a(n+3) = A008578(n+1) -a(n), a(0) = a(1) = a(2) = 0.

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A363473 Triangle read by rows: T(n, k) = k * prime(n - k + A061395(k)) for 1 < k <= n, and T(n, 1) = A008578(n).

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A370388 First appearance of prime_i in A369797 or 0 if no such appearance occurs, with p_0 being 1. (A008578 but with a different offset.)

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A000040 The prime numbers.

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A006530 Gpf(n): greatest prime dividing n, for n >= 2; a(1)=1.

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