cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 89 results. Next

A073437 Smallest x such that remainder Mod[A065855(x), A000720(x)]=n.

Original entry on oeis.org

4, 6, 8, 21, 22, 25, 26, 27, 30, 33, 66, 70, 77, 78, 82, 86, 87, 88, 92, 93, 94, 95, 96, 100, 116, 117, 118, 119, 120, 219, 220, 221, 222, 247, 248, 249, 250, 255, 256, 261, 262, 267, 268, 289, 290, 291, 292, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 323
Offset: 1

Views

Author

Labos Elemer, Jul 31 2002

Keywords

Examples

			Remainder 4 appears first as Mod[G[21],Pi[21]]= Mod[21-Pi[21]-1,Pi[21]]=Mod[21-8-1,8]=Mod[12,8]=4, so a(4)=21.

Crossrefs

Cf. A065855, A000720, A073435, A073436.

A000040 The prime numbers.

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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)

References

  • M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 2nd. ed., 2001; see p. 3.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
  • E. Bach and Jeffrey Shallit, Algorithmic Number Theory, I, Chaps. 8, 9.
  • D. M. Bressoud, Factorization and Primality Testing, Springer-Verlag NY 1989.
  • M. Cipolla, "La determinazione asintotica dell'n-mo numero primo.", Rend. d. R. Acc. di sc. fis. e mat. di Napoli, s. 3, VIII (1902), pp. 132-166.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 127-149.
  • R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 1.
  • 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.
  • J.-P. Delahaye, Merveilleux nombres premiers, Pour la Science-Belin Paris, 2000.
  • J.-P. Delahaye, Savoir si un nombre est premier: facile, Pour La Science, 303(1) 2003, pp. 98-102.
  • M. Dietzfelbinger, Primality Testing in Polynomial Time, Springer NY 2004.
  • M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 5.
  • J. Elie, "L'algorithme AKS", in 'Quadrature', No. 60, pp. 22-32, 2006 EDP-sciences, Les Ulis (France);
  • W. & F. Ellison, Prime Numbers, Hermann Paris 1985
  • T. Estermann, Introduction to Modern Prime Number Theory, Camb. Univ. Press, 1969.
  • J. M. Gandhi, Formulae for the nth prime. Proc. Washington State Univ. Conf. on Number Theory, 96-106. Wash. St. Univ., Pullman, Wash., 1971.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 77-78.
  • 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.
  • Peter Hilton and Jean Pedersen, A Mathematical Tapestry: Demonstrating the Beautiful Unity of Mathematics, Cambridge University Press, 2010, pp. (260-264).
  • H. D. Huskey, Derrick Henry Lehmer [1905-1991]. IEEE Ann. Hist. Comput. 17 (1995), no. 2, 64-68. Math. Rev. 96b:01035, cf. http://www.ams.org/mathscinet-getitem?mr=1336709
  • M. N. Huxley, The Distribution of Prime Numbers, Oxford Univ. Press, 1972.
  • D. S. Jandu, Prime Numbers And Factorization, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
  • E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea, NY, 1974.
  • D. H. Lehmer, The sieve problem for all-purpose computers. Math. Tables and Other Aids to Computation, Math. Tables and Other Aids to Computation, 7, (1953). 6-14. Math. Rev. 14:691e
  • D. N. Lehmer, "List of Prime Numbers from 1 to 10,006,721", Carnegie Institute, Washington, D.C. 1909.
  • W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Chap. 6.
  • H. Lifchitz, Table des nombres premiers de 0 à 20 millions (Tomes I & II), Albert Blanchard, Paris 1971.
  • R. F. Lukes, C. D. Patterson and H. C. Williams, Numerical sieving devices: their history and some applications. Nieuw Arch. Wisk. (4) 13 (1995), no. 1, 113-139. Math. Rev. 96m:11082, cf http://www.ams.org/mathscinet-getitem?mr=96m:11082
  • P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag NY 1995.
  • P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004.
  • H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhäuser Boston, Cambridge MA 1994.
  • B. Rittaud, "31415879. Ce nombre est-il premier?" ['Is this number prime?'], La Recherche, Vol. 361, pp. 70-73, Feb 15 2003, Paris.
  • D. Shanks, Solved and Unsolved Problems in Number Theory, 2nd. ed., Chelsea, 1978, Chap. 1.
  • 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).
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 107-119.
  • D. Wells, Prime Numbers: The Most Mysterious Figures In Math, J. Wiley NY 2005.
  • H. C. Williams and Jeffrey Shallit, Factoring integers before computers. Mathematics of Computation 1943-1993: a half-century of computational mathematics (Vancouver, BC, 1993), 481-531, Proc. Sympos. Appl. Math., 48, AMS, Providence, RI, 1994. Math. Rev. 95m:11143

Links

Crossrefs

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.

Programs

  • GAP
    A000040:=Filtered([1..10^5],IsPrime); # Muniru A Asiru, Sep 04 2017
  • Haskell
    -- 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) >;
  • Maple
    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, [$2..100]); # Peter Luschny, Feb 23 2018
  • Mathematica
    Prime[Range[60]]
  • Maxima
    A000040(n) := block(
if n = 1 then return(2),
return( next_prime(A000040(n-1)))
)$ /* recursive, to be replaced if possible - R. J. Mathar, Feb 27 2012 */
  • PARI
    {a(n) = if( n<1, 0, prime(n))};
  • 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): */
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
  • PARI
    forprime(p=2, 10^3, print1(p, ", ")) \\ Felix Fröhlich, Jun 30 2014
  • PARI
    primes(10^5) \\ Altug Alkan, Mar 26 2018
  • Python
    from sympy import primerange
print(list(primerange(2, 272))) # Michael S. Branicky, Apr 30 2022
  • Sage
    a = sloane.A000040
a.list(58)  # Jaap Spies, 2007
  • Sage
    prime_range(1, 300)  # Zerinvary Lajos, May 27 2009

Formula

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)

A000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n) to distinguish it from the number 3.14159...

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Partial sums of A010051 (characteristic function of primes). - Jeremy Gardiner, Aug 13 2002
pi(n) and prime(n) are inverse functions: a(A000040(n)) = n and A000040(n) is the least number m such that A000040(a(m)) = A000040(n). A000040(a(n)) = n if (and only if) n is prime. - Jonathan Sondow, Dec 27 2004
See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
A lower bound that gets better with larger N is that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i=0..T(sqrt(N))} A005867(i)/A002110(i). - Ben Paul Thurston, Aug 23 2010
Number of partitions of 2n into exactly two parts with the smallest part prime. - Wesley Ivan Hurt, Jul 20 2013
Equivalent to the Riemann hypothesis: abs(a(n) - li(n)) < sqrt(n)*log(n)/(8*Pi), for n >= 2657, where li(n) is the logarithmic integral (Lowell Schoenfeld). - Ilya Gutkovskiy, Jul 05 2016
The second Hardy-Littlewood conjecture, that pi(x) + pi(y) >= pi(x + y) for integers x and y with min{x, y} >= 2, is known to hold for (x, y) sufficiently large (Udrescu 1975). - Peter Luschny, Jan 12 2021

Examples

			There are 3 primes <= 6, namely 2, 3 and 5, so pi(6) = 3.

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
  • Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 8.
  • Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; p. 129.
  • Florian Cajori, A History of Mathematical Notations, Dover edition (2012), par. 409.
  • Richard Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 5.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorems 6, 7, 420.
  • G. J. O. Jameson, The Prime Number Theorem, Camb. Univ. Press, 2003. [See also the review by D. M. Bressoud (link below).]
  • Władysław Narkiewicz, The Development of Prime Number Theory, Springer-Verlag, 2000.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 132-133, 157-184.
  • József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.1. (For inequalities, etc.).
  • 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).
  • Gerald Tenenbaum and Michel Mendès France, Prime Numbers and Their Distribution, AMS Providence RI, 1999.
  • V. Udrescu, Some remarks concerning the conjecture pi(x + y) <= pi(x) + pi(y), Rev. Roumaine Math. Pures Appl. 20 (1975), 1201-1208.

Links

Crossrefs

Cf. A048989, A000040, A132090, A137588, A139328, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A143538, A036234, A033844, A034387, A034386, A179215, A010051, A212210-A212213, A233824, A056171, A304483.
Closely related:
A099802: Number of primes <= 2n.
A060715: Number of primes between n and 2n (exclusive).
A035250: Number of primes between n and 2n (inclusive).
A038107: Number of primes < n^2.
A014085: Number of primes between n^2 and (n+1)^2.
A007053: Number of primes <= 2^n.
A036378: Number of primes p between powers of 2, 2^n < p <= 2^(n+1).
A006880: Number of primes < 10^n.
A006879: Number of primes with n digits.
A033270: Number of odd primes <= n.
A065855: Number of composites <= n.
For lists of large values of a(n) see, e.g., A005669(n) = a(A002386(n)), A214935(n) = a(A205827(n)).
Related sequences:
Primes (p) and composites (c): A000040, A002808, 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.

Programs

  • Haskell
    a000720 n = a000720_list !! (n-1)
a000720_list = scanl1 (+) a010051_list  -- Reinhard Zumkeller, Sep 15 2011
  • Magma
    [ #PrimesUpTo(n): n in [1..200] ];  // Bruno Berselli, Jul 06 2011
  • Maple
    with(numtheory); A000720 := pi; [ seq(A000720(i),i=1..50) ];
  • Mathematica
    A000720[n_] := PrimePi[n]; Table[ A000720[n], {n, 1, 100} ]
Array[ PrimePi[ # ]&, 100 ]
Accumulate[Table[Boole[PrimeQ[n]],{n,100}]] (* Harvey P. Dale, Jan 17 2015 *)
  • PARI
    A000720=vector(100,n,omega(n!)) \\ For illustration only; better use A000720=primepi
  • PARI
    vector(300,j,primepi(j)) \\ Joerg Arndt, May 09 2008
  • Python
    from sympy import primepi
for n in range(1,100): print(primepi(n), end=', ') # Stefano Spezia, Nov 30 2018
  • Sage
    [prime_pi(n) for n in range(1, 79)]  # Zerinvary Lajos, Jun 06 2009

Formula

The prime number theorem gives the asymptotic expression a(n) ~ n/log(n).
For x > 1, pi(x) < (x / log x) * (1 + 3/(2 log x)). For x >= 59, pi(x) > (x / log x) * (1 + 1/(2 log x)). [Rosser and Schoenfeld]
For x >= 355991, pi(x) < (x / log(x)) * (1 + 1/log(x) + 2.51/(log(x))^2 ). For x >= 599, pi(x) > (x / log(x)) * (1 + 1/log(x)). [Dusart]
For x >= 55, x/(log(x) + 2) < pi(x) < x/(log(x) - 4). [Rosser]
For n > 1, A138194(n) <= a(n) <= A138195(n) (Tschebyscheff, 1850). - Reinhard Zumkeller, Mar 04 2008
For n >= 33, a(n) = 1 + Sum_{j=3..n} ((j-2)! - j*floor((j-2)!/j)) (Hardy and Wright); for n >= 1, a(n) = n - 1 + Sum_{j=2..n} (floor((2 - Sum_{i=1..j} (floor(j/i)-floor((j-1)/i)))/j)) (Ruiz and Sondow 2000). - Benoit Cloitre, Aug 31 2003
a(n) = A001221(A000142(n)). - Benoit Cloitre, Jun 03 2005
G.f.: Sum_{p prime} x^p/(1-x) = b(x)/(1-x), where b(x) is the g.f. for A010051. - Franklin T. Adams-Watters, Jun 15 2006
a(n) = A036234(n) - 1. - Jaroslav Krizek, Mar 23 2009
From Enrique Pérez Herrero, Jul 12 2010: (Start)
a(n) = Sum_{i=2..n} floor((i+1)/A000203(i)).
a(n) = Sum_{i=2..n} floor(A000010(n)/(i-1)).
a(n) = Sum_{i=2..n} floor(2/A000005(n)). (End)
Let pf(n) denote the set of prime factors of an integer n. Then a(n) = card(pf(n!/floor(n/2)!)). - Peter Luschny, Mar 13 2011
a(n) = -Sum_{p <= n} mu(p). - Wesley Ivan Hurt, Jan 04 2013
a(n) = (1/2)*Sum_{p <= n} (mu(p)*d(p)*sigma(p)*phi(p)) + sum_{p <= n} p^2. - Wesley Ivan Hurt, Jan 04 2013
a(1) = 0 and then, for all k >= 1, repeat k A001223(k) times. - Jean-Christophe Hervé, Oct 29 2013
a(n) = n/(log(n) - 1 - Sum_{k=1..m} A233824(k)/log(n)^k + O(1/log(n)^{m+1})) for m > 0. - Jonathan Sondow, Dec 19 2013
a(n) = A001221(A003418(n)). - Eric Desbiaux, May 01 2014
a(n) = Sum_{j=2..n} H(-sin^2 (Pi*(Gamma(j)+1)/j)) where H(x) is the Heaviside step function, taking H(0)=1. - Keshav Raghavan, Jun 18 2016
a(A014076(n)) = (1/2) * (A014076(n) + 1) - n + 1. - Christopher Heiling, Mar 03 2017
From Steven Foster Clark, Sep 25 2018: (Start)
a(n) = Sum_{m=1..n} A143519(m) * floor(n/m).
a(n) = Sum_{m=1..n} A001221(m) * A002321(floor(n/m)) where A002321() is the Mertens function.
a(n) = Sum_{m=1..n} |A143519(m)| * A002819(floor(n/m)) where A002819() is the Liouville Lambda summatory function and |x| is the absolute value of x.
a(n) = Sum_{m=1..n} A137851(m)/m * H(floor(n/m)) where H(n) = Sum_{m=1..n} 1/m is the harmonic number function.
a(n) = Sum_{m=1..log_2(n)} A008683(m) * A025528(floor(n^(1/m))) where A008683() is the Moebius mu function and A025528() is the prime-power counting function.
(End)
Sum_{k=2..n} 1/a(k) ~ (1/2) * log(n)^2 + O(log(n)) (de Koninck and Ivić, 1980). - Amiram Eldar, Mar 08 2021
a(n) ~ 1/(n^(1/n)-1). - Thomas Ordowski, Jan 30 2023
a(n) = Sum_{j=2..n} floor(((j - 1)! + 1)/j - floor((j - 1)!/j)) [Mináč, unpublished] (see Ribenboim, pp. 132-133). - Stefano Spezia, Apr 13 2025
a(n) = n - 1 - Sum_{k=2..floor(log_2(n))} pi_k(n), where pi_k(n) is the number of k-almost primes <= n. - Daniel Suteu, Aug 27 2025

Extensions

Additional links contributed by Lekraj Beedassy, Dec 23 2003
Edited by M. F. Hasler, Apr 27 2018 and (links recovered) Dec 21 2018

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

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.

Programs

  • Haskell
    a002808 n = a002808_list !! (n-1)
a002808_list = filter ((== 1) . a066247) [2..]
-- Reinhard Zumkeller, Feb 04 2012
  • Magma
    [n: n in [2..250] | not IsPrime(n)]; // G. C. Greubel, Feb 24 2024
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    A002808(n)=for(k=0,primepi(n),isprime(n++)&&k--);n \\ For illustration only: see below. - M. F. Hasler, Oct 31 2008
  • PARI
    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
  • PARI
    forcomposite(n=1, 1e2, print1(n, ", ")) \\ Felix Fröhlich, Aug 03 2014
  • PARI
    for(n=1, 1e3, if(bigomega(n) > 1, print1(n, ", "))) \\ Altug Alkan, Oct 14 2015
  • Python
    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
  • Python
    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
  • Python
    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
  • SageMath
    [n for n in (2..250) if not is_prime(n)] # G. C. Greubel, Feb 24 2024

Formula

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

Extensions

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

A373403 Length of the n-th maximal antirun of composite numbers differing by more than one.

Original entry on oeis.org

3, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Jun 05 2024

Keywords

Comments

This antirun ranges from A005381 (with 4 prepended) to A068780, with sum A373404.
An antirun of a sequence (in this case A002808) is an interval of positions such that consecutive terms differ by more than one.

Examples

			Row-lengths of:
   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

Links

Crossrefs

Functional neighbors: A005381, A027833 (partial sums A029707), A068780, A176246 (rest of A046933, firsts A073051), A373127, A373404, A373409.
A000040 lists the primes, differences A001223.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005117, A038664, A053767, A067970, A174965, A196274, A373400, A373573.

Programs

  • Mathematica
    Length/@Split[Select[Range[100],CompositeQ],#1+1!=#2&]//Most

Formula

a(2n) = 1.
a(2n - 1) = A196274(n) for n > 1.

A060715 Number of primes between n and 2n exclusive.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 12, 11, 12, 13, 13, 14, 14, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 15
Offset: 1

Views

Author

Lekraj Beedassy, Apr 25 2001

Keywords

Comments

See the additional references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
a(A060756(n)) = n and a(m) <> n for m < A060756(n). - Reinhard Zumkeller, Jan 08 2012
For prime n conjecturally a(n) = A226859(n). - Vladimir Shevelev, Jun 27 2013
The number of partitions of 2n+2 into exactly two parts where the first part is a prime strictly less than 2n+1. - Wesley Ivan Hurt, Aug 21 2013

Examples

			a(35)=8 since eight consecutive primes (37,41,43,47,53,59,61,67) are located between 35 and 70.

References

  • M. Aigner and C. M. Ziegler, Proofs from The Book, Chapter 2, Springer NY 2001.

Links

Crossrefs

Cf. A060756, A070046, A006992, A051501, A035250, A101909.
Cf. A000720, A014085, A104272, A143227, A128076.
Cf. A143223, A143224, A143225, A143226.
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.

Programs

  • Haskell
    a060715 n = sum $ map a010051 [n+1..2*n-1]  -- Reinhard Zumkeller, Jan 08 2012
  • Magma
    [0] cat [#PrimesInInterval(n+1, 2*n-1): n in [2..80]]; // Bruno Berselli, Sep 05 2012
  • Maple
    a := proc(n) local counter, i; counter := 0; from i from n+1 to 2*n-1 do if isprime(i) then counter := counter +1; fi; od; return counter; end:
with(numtheory); seq(pi(2*k-1)-pi(k),k=1..100); # Wesley Ivan Hurt, Aug 21 2013
  • Mathematica
    a[n_]:=PrimePi[2n-1]-PrimePi[n]; Table[a[n],{n,1,84}] (* Jean-François Alcover, Mar 20 2011 *)
  • PARI
    { for (n=1, 1000, write("b060715.txt", n, " ", primepi(2*n - 1) - primepi(n)); ) } \\ Harry J. Smith, Jul 10 2009
  • Python
    from sympy import primerange as pr
def A060715(n): return len(list(pr(n+1, 2*n))) # Karl-Heinz Hofmann, May 05 2022

Formula

a(n) = Sum_{k=1..n-1} A010051(n+k). - Reinhard Zumkeller, Dec 03 2009
a(n) = pi(2n-1) - pi(n). - Wesley Ivan Hurt, Aug 21 2013
a(n) = Sum_{k=(n^2-n+2)/2..(n^2+n-2)/2} A010051(A128076(k)). - Wesley Ivan Hurt, Jan 08 2022

Extensions

Corrected by Dug Eichelberger (dug(AT)mit.edu), Jun 04 2001
More terms from Larry Reeves (larryr(AT)acm.org), Jun 05 2001

A062298 Number of nonprimes <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 16, 17, 18, 19, 19, 20, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 28, 29, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 37, 38, 39, 40, 41, 42, 42, 43, 43, 44, 45, 46, 47, 48, 48, 49, 50, 51, 51, 52, 52, 53
Offset: 1

Views

Author

Amarnath Murthy, Jun 19 2001

Keywords

Comments

a(n) = n - A000720(n). This is asymptotic to n - Li(n). Note that a(n) + A095117(n) = 2*n. - Jonathan Vos Post, Nov 22 2004
Same as number of primes between n and prime(n+1) and between n and prime(n)+1 (end points excluded); n prime -> a(n)=a(n-1), n composite-> a(n)=1+a(n-1). - David James Sycamore, Jul 23 2018
There exists at least one prime number between a(n) and n for n >= 3 (see the paper by Ya-Ping Lu attached in the links). - Ya-Ping Lu, Nov 27 2020

Examples

			a(19) = 11 as there are 8 primes up to 19 (inclusive).

Links

Crossrefs

Cf. A000720, A101203, A010051, A065855.

Programs

  • Haskell
    a062298 n = a062298_list !! (n-1)
a062298_list = scanl1 (+) $ map (1 -) a010051_list
-- Reinhard Zumkeller, Oct 10 2013
  • Magma
    [n - #PrimesUpTo(n): n in [1..100]]; // Vincenzo Librandi, Aug 05 2015
  • Maple
    NumComposites := proc(N::posint) local count, i:count := 0:for i from 1 to N do if not isprime(i) then count := count + 1 fi:od: count;end:seq(NumComposites(binomial(k+1,k)), k=0..73); # Zerinvary Lajos, May 26 2008
A062298 := proc(n) n-numtheory[pi](n) ; end: seq(A062298(n),n=1..120) ; # R. J. Mathar, Sep 27 2009
  • Mathematica
    Table[n-PrimePi[n],{n,80}] (* Harvey P. Dale, May 10 2012 *)
Accumulate[Table[If[PrimeQ[n],0,1],{n,100}]] (* Harvey P. Dale, Feb 15 2017 *)
  • PARI
    a(n) = n-primepi(n); \\ Harry J. Smith, Aug 04 2009
  • Python
    from sympy import primepi
print([n - primepi(n) for n in range(1, 101)]) # Indranil Ghosh, Mar 29 2017

Formula

a(n) = n - A000720(n).
a(n) = 1 + A065855(n). - David James Sycamore, Jul 23 2018

Extensions

Corrected and extended by Vladeta Jovovic, Jun 22 2001

A065890 Number of composites less than the n-th prime.

Original entry on oeis.org

0, 0, 1, 2, 5, 6, 9, 10, 13, 18, 19, 24, 27, 28, 31, 36, 41, 42, 47, 50, 51, 56, 59, 64, 71, 74, 75, 78, 79, 82, 95, 98, 103, 104, 113, 114, 119, 124, 127, 132, 137, 138, 147, 148, 151, 152, 163, 174, 177, 178, 181, 186, 187, 196, 201, 206, 211, 212, 217, 220, 221
Offset: 1

Views

Author

Labos Elemer and Robert G. Wilson v, Nov 28 2001

Keywords

Comments

First differences form A046933, which requires that for this sequence the parity of successive terms alternates.

Examples

			a(25) = 71 since prime(25) = 97 is the 25th prime and 96 is the 71st composite number in A002808.

Links

Crossrefs

Cf. A000040, A014689, A014692, A002808, A018252, A065855, A046933.

Programs

  • Magma
    [NthPrime(n)-n-1: n in [1..65]]; // Vincenzo Librandi, Aug 15 2015
  • Mathematica
    CompositePi[n_Integer] := (n - PrimePi[n] - 1); Table[ CompositePi[ Prime[n]], {n, 1, 75} ]
  • PARI
    a(n) = { prime(n) - n - 1 } \\ Harry J. Smith, Nov 03 2009
  • Python
    from sympy import prime
def A065890(n): return prime(n)-n-1 # Chai Wah Wu, Oct 11 2024

Formula

a(n) = A065855(A000040(n)).
a(n) = A000040(n)-n-1 = A014689(n)-1 = A014692(n)-2.

A035250 Number of primes between n and 2n (inclusive).

Original entry on oeis.org

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

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Author

Erich Friedman

Keywords

Comments

By Bertrand's Postulate (proved by Chebyshev), there is always a prime between n and 2n, i.e., a(n) is positive for all n.
The number of primes in the interval [n,2*n) is the same sequence as this, except that a(1) = 0. - N. J. A. Sloane, Oct 18 2024
The smallest and largest primes between n and 2n inclusive are A007918 and A060308 respectively. - Lekraj Beedassy, Jan 01 2007
The number of partitions of 2n into exactly two parts with first part prime, n > 1. - Wesley Ivan Hurt, Jun 15 2013

Examples

			The primes between n = 13 and 2n = 26, inclusive, are 13, 17, 19, 23; so a(13) = 4.
a(5) = 2, since 2(5) = 10 has 5 partitions into exactly two parts: (9,1),(8,2),(7,3),(6,4),(5,5).  Two primes are among the first parts: 7 and 5.

References

  • Aigner, M. and Ziegler, G. Proofs from The Book (2nd edition). Springer-Verlag, 2001.

Links

Crossrefs

Cf. A073837, A073838, A099802, A060715.
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.

Programs

Formula

a(n) = A000720(2*n) - A000720(n-1); a(n) <= A179211(n). - Reinhard Zumkeller, Jul 05 2010
a(A059316(n)) = n and a(m) <> n for m < A059316(n). - Reinhard Zumkeller, Jan 08 2012
a(n) = sum(A010051(k): k=n..2*n). [Reinhard Zumkeller, Jan 08 2012]
a(n) = pi(2n) - pi(n-1). [Wesley Ivan Hurt, Jun 15 2013]

A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4, 10, 2, 3, 6, 4, 2, 2, 7, 2, 6, 3
Offset: 1

Views

Author

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

Keywords

Comments

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012
Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012
Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - Richard R. Forberg, Jul 28 2016
All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016
From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These begin:
5
7 11
13 17
19 23 29
31 37 41
43 47 53 59
61 67 71
73 79 83 89 97 101
(End)

Links

Crossrefs

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012
Positions of first appearances are A373401, sorted A373402.
Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A025157, A038664, A071148, A176246, A350842, A373406.

Programs

  • Maple
    A027833 := proc(n)
    local plow,phigh ;
    phigh := A001359(n+1) ;
    plow := A001359(n) ;
    numtheory[pi](phigh)-numtheory[pi](plow) ;
end proc:
seq(A027833(n),n=1..100) ; # R. J. Mathar, Jan 20 2025
  • Mathematica
    Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* Harvey P. Dale, Nov 17 2018 *)
Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)
  • PARI
    n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ Charles R Greathouse IV, Aug 01 2016
  • Sage
    def A027833(n) :
   a = [ ]
   st = 2
   for i in (3..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i-st)
         st = i
   return(a)
A027833(496) # Jani Melik, May 15 2014
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