A073838 -id:A073838 - cp's OEIS frontend

A034386 Primorial numbers (second definition): n# = product of primes <= n.

1, 1, 2, 6, 6, 30, 30, 210, 210, 210, 210, 2310, 2310, 30030, 30030, 30030, 30030, 510510, 510510, 9699690, 9699690, 9699690, 9699690, 223092870, 223092870, 223092870, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130

Offset: 0

N. J. A. Sloane

Squarefree kernel of both n! and lcm(1, 2, 3, ..., n).
a(n) = lcm(core(1), core(2), core(3), ..., core(n)) where core(x) denotes the squarefree part of x, the smallest integer such that x*core(x) is a square. - Benoit Cloitre, May 31 2002
The sequence can also be obtained by taking a(1) = 1 and then multiplying the previous term by n if n is coprime to the previous term a(n-1) and taking a(n) = a(n-1) otherwise. - Amarnath Murthy, Oct 30 2002; corrected by Franklin T. Adams-Watters, Dec 13 2006

			a(5) = a(6) = 2*3*5 = 30;
a(7) = 2*3*5*7 = 210.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, p. 14, "n?".
József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section VII.35, p. 268.

Alois P. Heinz, Table of n, a(n) for n = 0..2370 (first 401 terms from T. D. Noe)
Jens Askgaard, On the additive period length of the Sprague-Grundy function of certain Nim-like games, arXiv:1902.06299 [math.CO], 2019.
Klaus Dohmen and Martin Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv:1404.5480 [math.CO], 2014.
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), 64-94.
Eric Weisstein's World of Mathematics, Primorial.

Cf. A002110, A057872, A249270.
Cf. A073838, A034387. - Reinhard Zumkeller, Jul 05 2010
The following fractions are all related to each other: Sum 1/n: A001008/A002805, Sum 1/prime(n): A024451/A002110 and A106830/A034386, Sum 1/nonprime(n): A282511/A282512, Sum 1/composite(n): A250133/A296358.

[n eq 0 select 1 else LCM(PrimesInInterval(1, n)) : n in [0..50]]; // G. C. Greubel, Jul 21 2023

A034386 := n -> mul(k,k=select(isprime,[$1..n])); # Peter Luschny, Jun 19 2009
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      `if`(isprime(n), n, 1)*a(n-1))
    end:
seq(a(n), n=0..36);  # Alois P. Heinz, Nov 26 2020

q[x_]:=Apply[Times,Table[Prime[w],{w,1,PrimePi[x]}]]; Table[q[w],{w,1,30}]
With[{pr=FoldList[Times,1,Prime[Range[20]]]},Table[pr[[PrimePi[n]+1]],{n,0,40}]] (* Harvey P. Dale, Apr 05 2012 *)
Table[ResourceFunction["Primorial"][i], {i,1,40}] (* Navvye Anand, May 22 2024 *)

a(n)=my(v=primes(primepi(n)));prod(i=1,#v,v[i]) \\ Charles R Greathouse IV, Jun 15 2011

a(n)=lcm(primes([2,n])) \\ Jeppe Stig Nielsen, Mar 10 2019

from sympy import primorial
def A034386(n): return 1 if n == 0 else primorial(n,nth=False) # Chai Wah Wu, Jan 11 2022

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
[sharp_primorial(n) for n in (0..30)] # Giuseppe Coppoletta, Jan 26 2015

a(n) = n# = A002110(A000720(n)) = A007947(A003418(n)) = A007947(A000142(n)).
Asymptotic expression for a(n): exp((1 + o(1)) * n) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, log(a(n)) < 1.01624*n. [Rosser and Schoenfeld, 1962; Sándor et al., 2005] - N. J. A. Sloane, Apr 04 2017
a(n) <= A179215(n). - Reinhard Zumkeller, Jul 05 2010
a(n) = lcm(A006530(n), a(n-1)). - Jon Maiga, Nov 10 2018
Sum_{n>=0} 1/a(n) = A249270. - Amiram Eldar, Nov 08 2020

Offset changed and initial term added by Arkadiusz Wesolowski, Jun 04 2011

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

Erich Friedman

nonn

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

			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.

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

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
International Mathematics Olympiad, Proof of Bertrand's Postulate [Via Wayback Machine]

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.

a035250 n = sum $ map a010051 [n..2*n] -- Reinhard Zumkeller, Jan 08 2012

[#PrimesInInterval(n, 2*n): n in [1..80]]; // Bruno Berselli, Sep 05 2012

with(numtheory): A035250:=n->pi(2*n)-pi(n-1): seq(A035250(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

f[n_] := PrimePi[2n] - PrimePi[n - 1]; Array[f, 76] (* Robert G. Wilson v, Dec 23 2012 *)

a(n)=primepi(2*n)-primepi(n-1) \\ Charles R Greathouse IV, Jul 01 2013

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]

A126804 a(n) = (2n)! / (n-1)!.

Original entry on oeis.org

2, 24, 360, 6720, 151200, 3991680, 121080960, 4151347200, 158789030400, 6704425728000, 309744468633600, 15543540607795200, 841941782922240000, 48962152914554880000, 3042648073975910400000, 201220459292273541120000, 14110584707870682071040000

Offset: 1

Jonathan R. Love (japanada11(AT)yahoo.ca), Feb 22 2007

Old name was "Multiplying n X n integers above n".
a(n) = 2*A001814(n). - Zerinvary Lajos, May 03 2007
A179214(n) <= a(n). - Reinhard Zumkeller, Jul 05 2010
Product of the numbers from n to 2n. - Wesley Ivan Hurt, Dec 14 2015

			a(5) = 151200 because five digits above 5: (6, 7, 8, 9, 10), multiplied by five equals 5*(6*7*8*9*10) = 151200.

Robert Israel, Table of n, a(n) for n = 1..360
Elliot J. Carr and Matthew J. Simpson, Accurate and efficient calculation of finite response times for groundwater flow, arXiv:1707.06331 [physics.flu-dyn], 2017, p. 11.

Cf. A045943, A073838. - Reinhard Zumkeller, Jul 05 2010

Cf. A001814, A179214.

[Factorial(2*n)/Factorial(n-1) : n in [1..20]]; // Wesley Ivan Hurt, Dec 14 2015

a:=n->sum((count(Permutation(2*n+2),size=n+1)),j=0..n): seq(a(n), n=0..15); # Zerinvary Lajos, May 03 2007
seq(mul((n+k), k=0..n), n=1..16); # Zerinvary Lajos, Sep 21 2007
with(combstruct):with(combinat) :bin := {B=Union(Z,Prod(B,B))}: seq (count([B,bin,labeled],size=n)*(n-1), n=2..17); # Zerinvary Lajos, Dec 05 2007

Table[Pochhammer[n, n + 1], {n, 17}] (* Arkadiusz Wesolowski, Aug 13 2012 *)
Table[(2 n)!/(n - 1)!, {n, 20}] (* Wesley Ivan Hurt, Dec 14 2015 *)

a(n) = prod(k=n, 2*n, k); \\ Michel Marcus, Dec 15 2015

x='x+O('x^99); Vec(serlaplace(2*x/(1-4*x)^(3/2))) \\ Altug Alkan, Mar 11 2018

a(n) = (2n)! / (n-1)!.
a(n) = Product_{i=n..2n} i. - Wesley Ivan Hurt, Dec 14 2015
From Robert Israel, Dec 15 2015: (Start)
a(n) = (2*n*(2*n-1)/(n-1))*a(n-1).
E.g.f.: 2*x/(1-4*x)^(3/2). (End)
a(n) = Pochhammer(n,n+1). - Pedro Caceres, Mar 10 2018

New name from Wesley Ivan Hurt, Dec 15 2015

A073837 Sum of primes p satisfying n <= p <= 2n.

Original entry on oeis.org

2, 5, 8, 12, 12, 18, 31, 24, 41, 60, 60, 72, 72, 59, 88, 119, 119, 102, 139, 120, 161, 204, 204, 228, 228, 228, 281, 281, 281, 311, 372, 341, 341, 408, 408, 479, 552, 515, 515, 594, 594, 636, 636, 593, 682, 682, 682, 635, 732, 732, 833, 936, 936, 990, 1099, 1099

Offset: 1

Amarnath Murthy, Aug 12 2002

nonn

a(n) = A034387(2*n) - A034387(n-1); a(n) <= A179213(n). [Reinhard Zumkeller, Jul 05 2010]

			a(7) = 31 = 7+11+13 (sum of primes between 7 and 14).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A073838.

for n from 1 to 150 do l := 0:for j from n to 2*n do if isprime(j) then l := l+j:fi:od:a[n] := l:od:seq(a[j],j=1..150);

Table[Total[Select[Range[n, 2 n], PrimeQ]], {n, 56}] (* Jayanta Basu, Aug 12 2013 *)

PARI
```
a(n)=sum(i=n,2*n,i*isprime(i))
```

More terms from Sascha Kurz and Benoit Cloitre, Aug 14 2002

A179214 Product of squarefree numbers between n and 2*n (inclusive).

Original entry on oeis.org

2, 6, 90, 210, 2100, 4620, 140140, 300300, 5105100, 96996900, 4481256780, 9369900540, 243617414040, 18739801080, 1164544781400, 2406725881560, 2700346439110320, 5559536786403600, 7816708721683461600

Offset: 1

Reinhard Zumkeller, Jul 05 2010

nonn

A073838(n) <= a(n) <= A126804(n);

a(n) = A179215(2*n)/A179215(n-1).

			a(10) = 10*11*13*14*15*17*19 = 96996900;
a(11) = 11*13*14*15*17*19*21*22 = 4481256780;
a(12) = 13*14*15*17*19*21*22*23 = 9369900540.

Cf. A179211, A179213, A005117.

a(n) = PROD(k^A008966(k): n <= k <= 2*n).

Example corrected by Reinhard Zumkeller, Jul 19 2010

A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 6, 3, 1, 1, 30, 15, 5, 5, 1, 30, 15, 5, 5, 1, 1, 210, 105, 35, 35, 7, 7, 1, 210, 105, 35, 35, 7, 7, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 210, 105, 35, 35, 7, 7, 1, 1, 1, 1, 2310, 1155, 385, 385, 77, 77, 11, 11, 11, 11, 1, 2310, 1155, 385, 385

Offset: 1

Arkadiusz Wesolowski, Nov 27 2011

Row sums give A201156.
Central terms give A068111: T(2*n-1,n) = A068111(n).
T(n,1) = A034386(n).
T(n,n-1) = A089026(n) for n > 1.
T(n,n) = A000012(n).
Let n > 1 and p = A000040(n). Then T(p,p-1) = T(p+1,p-1) = p.
T(2*n-1,n-1) = A073838(n) for n > 1.
T(2*n,n+1) = A144186(n).

			1:                                   1
2:                               2       1
3:                           6       3       1
4:                       6       3       1       1
5:                   30      15      5       5       1
6:               30      15      5       5       1       1
7:           210     105     35      35      7       7       1
8:       210     105     35      35      7       7       1       1
9:   210     105     35      35      7       7       1       1       1

Arkadiusz Wesolowski, Rows n = 1..141 of triangle, flattened

Cf. A034386.

lst = {}; Do[AppendTo[lst, Product[Prime[i], {i, PrimePi[n]}]/Product[Prime[i], {i, PrimePi[k]}]], {n, 12}, {k, n}]; lst (* Arkadiusz Wesolowski, Dec 02 2011 *)

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A034386 Primorial numbers (second definition): n# = product of primes <= n.

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

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A126804 a(n) = (2n)! / (n-1)!.

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A073837 Sum of primes p satisfying n <= p <= 2n.

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A179214 Product of squarefree numbers between n and 2*n (inclusive).

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A201146 Triangle read by rows: T(n,k) = (n#)/(k#), 1 <= k <= n.

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