A233824 -id:A233824 - cp's OEIS frontend

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

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

N. J. A. Sloane

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

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

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.

Daniel Forgues, Table of n, pi(n) for n = 1..100000 (first 20000 terms from N. J. A. Sloane; see below for links with 823852 terms (Verma) and more (Caldwell))
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].
Christian Axler, Über die Primzahl-Zählfunktion, die n-te Primzahl und verallgemeinerte Ramanujan-Primzahlen, Ph.D. thesis 2013, in German, English summary.
Paul T. Bateman and Harold G. Diamond, A Hundred Years of Prime Numbers, Amer. Math. Month., Vol. 103, No. 9 (Nov. 1996), pp. 729-741, MAA Washington DC.
Steve Bennett, The role of Riemann's zeta function in the analytic proof of the Prime Number Theorem, 2004.
Claudio Bonanno and Mirko S. Mega, Toward a dynamical model for prime numbers, Chaos, Solitons & Fractals, Vol. 20, No. 1 (2004), pp. 107-118; arXiv preprint, arXiv:cond-mat/0309251, 2003.
David M. Bressoud, Review of "The Prime Number Theorem" by G. J. O. Jameson, MAA Reviews, 2005. - from _N. J. A. Sloane_, Dec 29 2018
D. M. Bressoud and Stan Wagon, Computational Number Theory: Basic Algorithms, Springer/Key, 2000 (with a Mathematica package for computational number theory).
Ben Brubaker, The Prime Number Theorem.
C. K. Caldwell, The Prime Glossary: Prime number theorem.
W. W. L. Chen, Distribution of Prime Numbers.
Jean-Marie de Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See Chapter 9, p. 231.
Marc Deléglise, Computation of large values of pi(x), 1996.
Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers, Thèse, Université de Limoges, France (1998).
Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation, Vo. 68, No. 225 (1999), pp. 411-415.
Encyclopedia Britannica, The Prime Number Theorem [web.archive.org's copy of a no longer available personal copy of the Encyclopedia's article]
R. Gray and J. D. Mitchell, Largest subsemigroups of the full transformation monoid, Discrete Math., 308 (2008), 4801-4810.
G. H. Hardy and J. E. Littlewood, Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes, Acta Mathematica, Vol. 41 (1916), pp. 119-196.
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
Mehdi Hassani, Approximation of pi(x) by Psi(x), J. Inequ. Pure Appl. Math., Vol. 7, No. 1 (2006), Article #7.
Y.-C. Kim, Note on the Prime Number Theorem, arXiv:math/0502062 [math.NT], 2005.
Tzanio V. Kolev, On the number of Prime Numbers less than a Given Quantity, 2000.
Angel V. Kumchev, The Distribution of Prime Numbers, 2005.
J. C. Lagarias, V. S. Miller and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., Vol. 44, No. 170 (1985), pp. 537-560.
Jeffrey C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, Journal of Algorithms, Vol. 8, No. 2 (1987), pp. 173-191; alternative link.
John Lorch, The Distribution of Primes, B.S. Undergraduate Mathematics Exchange, Vol. 3, No. 1 (Fall 2005).
Nathan McKenzie, Computing the Prime Counting Function with Linnik's Identity, personal blog, March 24, 2011.
Murat Baris Paksoy, Derived Ramanujan primes: R'_n, arXiv:1210.6991 [math.NT], 2012.
Bent E. Petersen, Prime Number Theorem, Seminar Lecture Note, 1996; version 2002-05-14.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos, personal website, 2001 (?); Illustration of initial terms: Divisors and pi(x).
Bernhard Riemann, On the Number of Prime Numbers, 1859, last page (various transcripts)
J. Barkley Rosser, Explicit Bounds for Some Functions of Prime Numbers, American Journal of Mathematics, Vol. 63, No. 1 (1941), pp. 211-232.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 64-94.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy).
Sebastian Martin Ruiz and Jonathan Sondow, Formulas for pi(n) and the n-th prime, arXiv:math/0210312 [math.NT], 2002, 2014.
Jeffrey Shallit, Bibliography on calculation of pi(x).
Jonathan Sondow, Ramanujan Primes and Bertrand's Postulate, The American Mathematical Monthly, Vol. 116, No. 7 (2009), pp. 630-635; arXiv preprint, arXiv:0907.5232 [math.NT], 2009-2010.
Igor Turkanov, The prime counting function, arXiv:1603.02914 [math.NT], 2016.
Gaurav Verma and Srujan Sapkal, Table of n, pi(n) for n = 1..823852.
Matthew R. Watkins, The distribution of Prime Numbers.
Matthew R. Watkins, The prime number theorem (some references).
Eric Weisstein's World of Mathematics, Prime Counting Function.
Wikipedia, Prime number theorem.
Wikipedia, Prime-counting function.
C. P. Willans, On formulae for the nth prime, Math. Gazette 48 (1964), 413-415.
Marek Wolf, Applications of Statistical Mechanics in Number Theory, Physica A, vol. 274, no. 2, 1999, pp. 149-157; 1999 preprint.
Wolfram Research, First 50 values of pi(n).
Index entries for "core" sequences

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.

a000720 n = a000720_list !! (n-1)
a000720_list = scanl1 (+) a010051_list  -- Reinhard Zumkeller, Sep 15 2011

[ #PrimesUpTo(n): n in [1..200] ];  // Bruno Berselli, Jul 06 2011

with(numtheory); A000720 := pi; [ seq(A000720(i),i=1..50) ];

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 *)

A000720=vector(100,n,omega(n!)) \\ For illustration only; better use A000720=primepi

vector(300,j,primepi(j)) \\ Joerg Arndt, May 09 2008

from sympy import primepi
for n in range(1,100): print(primepi(n), end=', ') # Stefano Spezia, Nov 30 2018

[prime_pi(n) for n in range(1, 79)]  # Zerinvary Lajos, Jun 06 2009

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

Additional links contributed by Lekraj Beedassy, Dec 23 2003

Edited by M. F. Hasler, Apr 27 2018 and (links recovered) Dec 21 2018

A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

Original entry on oeis.org

1, 1, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605, 46066653228356851631, 1018705098450570562877

Offset: 0

N. J. A. Sloane

Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5].
Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3].
Related to number of subgroups of index n-1 in free group of rank 2 (i.e., maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.
Also the left border of triangle A144107, with row sums = n!. - Gary W. Adamson, Sep 11 2008
Hankel transform is A059332. Hankel transform of aerated sequence is A137704(n+1). - Paul Barry, Oct 07 2008
For every n, a(n+1) is also the moment of order n for the probability density function rho(x) = exp(x)/(Ei(1,-x)*(Ei(1,-x) + 2*I*Pi)) on the interval 0..infinity, with Ei the exponential-integral function. - Groux Roland, Jan 16 2009
Also (apparently), a(n+1) is the number of rooted hypermaps with n darts on a surface of any genus (see Walsh 2012). - N. J. A. Sloane, Aug 01 2012
Also recurrent sequence A233824 (for n > 0) in Panaitopol's formula for pi(x), the number of primes <= x. - Jonathan Sondow, Dec 19 2013
Also the number of mobiles (cyclic rooted trees) with an arrow from each internal vertex to a descendant of that vertex. - Brad R. Jones, Sep 12 2014
Up to sign, Möbius numbers of the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - F. Chapoton, Apr 29 2015
Also, a(n) is the number of distinct leaf matrices of complete non-ambiguous trees of size n. - Daniel Chen, Oct 23 2022

			G.f. = 1 + x + x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 461*x^6 + 3447*x^7 + 29093*x^8 + ...
From _Peter Luschny_, Aug 03 2022: (Start)
A permutation p in [n] (where n >= 0) is reducible if there exists an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation at i.
Examples:
* () is not reducible since there is no index i which splits (). (=> a(0) = 1)
* (1) is not reducible since there is no index i which splits (1). (=> a(1) = 1)
* (1, 2) is reducible since index 1 splits (1, 2) as p(1) < p(2).
* (2, 1) is not reducible since at the only potential splitting point i = 1 we have p(1) > p(2). (=> a(2) = 1)
* For n = 3 we have (1, 2, 3), (1, 3, 2), and (2, 1, 3) are reducible and (2, 3, 1), (3, 1, 2), and (3, 2, 1) are irreducible. (End)

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Seiichi Manyama, Table of n, a(n) for n = 0..449 (first 102 terms from T. D. Noe)
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R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
Richard J. Martin and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions (2008); arXiv:0806.3682 [math.CO]. Discrete Math. 310 (2010), no. 24, 3584-3606.
P. Ossona de Mendez and P. Rosenstiehl, Transitivity and connectivity of permutations, Combinatorica, 24 (No. 3, 2004), 487-501.
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See A167894 for another version.
Bisections give A272656, A272657.
Row sums of A111184 and A089949.
Leading diagonal of A059438. A diagonal of A263484.
Cf. A051296, A084938, A074664, A113869, A144107.
Cf. A260503, A259472, A261214, A261239, A261253, A261254.
Cf. A090238, A000698, A356291 (reducible permutations).
Column k=0 of A370380 and A370381 (without pair of initial terms and with different offset).

INVERTi([seq(n!,n=1..20)]);
A003319 := proc(n) option remember; n! - add((n-j)!*A003319(j), j=1..n-1) end;
[seq(A003319(n), n=0..50)]; # N. J. A. Sloane, Dec 28 2011
series(2 - 1/hypergeom([1,1], [], x), x=0,50); # Mark van Hoeij, Apr 18 2013

a[n_] := a[n] = n! - Sum[k!*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 11 2011, after given formula *)
CoefficientList[Assuming[Element[x,Reals],Series[2-E^(1/x)* x/ExpIntegralEi[1/x],{x,0,20}]],x] (* Vaclav Kotesovec, Mar 07 2014 *)
a[ n_] := If[ n < 2, 1, a[n] = (n - 2) a[n - 1] + Sum[ a[k] a[n - k], {k, n - 1}]]; (* Michael Somos, Feb 23 2015 *)
Table[SeriesCoefficient[1 + x/(1 + ContinuedFractionK[-Floor[(k + 2)/2]*x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 29 2017 *)

{a(n) = my(A); if( n<1, 1, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (k - 2) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */

{if(n<1,1,a(n)=local(A=x);for(i=1,n,A=x-x*A+A^2+x^2*A' +x*O(x^n));polcoeff(A,n))} /* Paul D. Hanna, Jul 30 2011 */

def A003319_list(len):
    R, C = [1], [1] + [0] * (len - 1)
    for n in range(1, len):
        for k in range(n, 0, -1):
            C[k] = C[k - 1] * k
        C[0] = -sum(C[k] for k in range(1, n + 1))
        R.append(-C[0])
    return R
print(A003319_list(21))  # Peter Luschny, Feb 19 2016

G.f.: 2 - 1/Sum_{k>=0} k!*x^k.
Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k) [Bowen, 1976].
Also coefficients in the divergent series expansion log Sum_{n>=0} n!*x^n = Sum_{n>=1} a(n+1)*x^n/n [Bowen, 1976].
a(n) = (-1)^(n-1) * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}.
INVERTi transform of factorial numbers, A000142 starting from n=1. - Antti Karttunen, May 30 2003
Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle A089949. - Philippe Deléham, Dec 30 2003
a(n+1) = Sum_{k=0..n} A089949(n,k). - Philippe Deléham, Oct 16 2006
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ). - Paul D. Hanna, Sep 19 2007
G.f.: 1+x/(1-x/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-...)))))))) (continued fraction). - Paul Barry, Oct 07 2008
a(n) = -Sum_{i=0..n} (-1)^i*A090238(n, i) for n > 0. - Peter Luschny, Mar 13 2009
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = upper left term in M^(n-1), M = triangle A128175 as an infinite square production matrix (deleting the first "1"); as follows:
   1,  1,  0,  0,  0,  0, ...
   2,  2,  1,  0,  0,  0, ...
   4,  4,  3,  1,  0,  0, ...
   8,  8,  7,  4,  1,  0, ...
  16, 16, 15, 11,  5,  1, ...
  ... (End)
O.g.f. satisfies: A(x) = x - x*A(x) + A(x)^2 + x^2*A'(x). - Paul D. Hanna, Jul 30 2011
From Sergei N. Gladkovskii, Jun 24 2012: (Start)
Let A(x) be the g.f.; then
A(x) = 1/Q(0), where Q(k) = x + 1 + x*k - (k+2)*x/Q(k+1).
A(x) = (1-1/U(0))/x, when U(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/U(k+1))). (End)
From Sergei N. Gladkovskii, Aug 03 2013: (Start)
Continued fractions:
G.f.: 1 - G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))).
G.f.: (x/2)*G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1/2) + 1/G(k+1))).
G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1)).
G.f.: 1 - 1/G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1)))).
G.f.: x*W(0), where W(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+2)/(x*(k+2) - 1/W(k+1)))).
(End)
a(n) = A233824(n-1) if n > 0. (Proof. Set b(n) = A233824(n), so that b(n) = n*n! - Sum_{k=1..n-1} k!*b(n-k). To get a(n+1) = b(n) for n >= 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.) - Jonathan Sondow, Dec 19 2013
a(n) ~ n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10), for coefficients see A260503. - Vaclav Kotesovec, Jul 27 2015
For n>0, a(n) = (A059439(n) - A259472(n))/2. - Vaclav Kotesovec, Aug 03 2015
From Peter Bala, May 23 2017: (Start)
G.f.: 1 + x/(1 + x - 2*x/(1 + 2*x - 3*x/(1 + 3*x - 4*x/(1 + 4*x - ...)))). Cf. A000698.
G.f.: 1/(1 - x/(1 + x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ...))))))))). (End)
Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - Mikhail Kurkov, Apr 26 2024

More terms from Michael Somos, Jan 26 2000
Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002
Added a(0)=0 (some of the formulas may now need adjusting). - N. J. A. Sloane, Sep 12 2012
Edited and set a(0) = 1 by Peter Luschny, Aug 03 2022

For large n, the ratio prime(n)/a(n) tends to e (very slowly). This was conjectured by Anton Vrba in 2010 (see Rivera, 2010) and proved by Sandor and Verroken (2011). Tighter bounds and asymptotics for a(n) are proved in the note "On the geometric mean of the first n primes" (2016) (see links). Better formulas prime(n)/a(n) ~ exp(1 + several terms A233824(k) / log^k(prime(n))) exist for larger n; see examples in the formula section. - Alexei Kourbatov, Feb 27 2016.

cp's OEIS Frontend

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

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A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

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A062049 Integer part of geometric mean of first n primes.

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A272428 Expansion of the continued fraction x/(1-pi(2)*x/(1-pi(3)*x/(1-pi(4)*x/(1-...)))).

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A330436 a(n) = n * n!! - Sum_{k=1..n-1} k!! * a(n-k).

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A272428 Expansion of the continued fraction x/(1-pi(2)x/(1-pi(3)x/(1-pi(4)*x/(1-...)))).