A057872 -id:A057872 - 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

A035158 Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 5, 5, 5, 5, 7, 7, 10, 10, 10, 10, 13, 13, 16, 16, 16, 16, 19, 19, 19, 19, 19, 19, 22, 22, 26, 26, 26, 26, 26, 26, 29, 29, 29, 29, 33, 33, 37, 37, 37, 37, 40, 40, 40, 40, 40, 40, 44, 44, 44, 44, 44, 44, 49, 49, 53, 53, 53, 53, 53, 53, 57, 57, 57, 57, 61, 61, 65, 65, 65, 65

Offset: 1

N. J. A. Sloane, Oct 02 2008

nonn

The old entry with this sequence number was a duplicate of A002325.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, see Chap. 22.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section VII.35. (For inequalities, etc.)

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)
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x), Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29 (1975), 243-269.
J. Barkley Rosser and Lowell Schoenfeld, Sharper bounds for the Chebyshev functions theta (x) and psi (x). II. Math. Comp. 30 (1976), number 134, 337-360.
J. Barkley Rosser and Lowell Schoenfeld, Corrigendum: "Sharper bounds for the Chebyshev functions theta (x) and psi (x). II" (Math. Comput. 30 (1976), number 134, 337-360), Math. Comp. 30 (1976), number 136, 900.

Cf. A057872, A083535, A016040 (records), A000040 (places of records)

(Maple for A035158, A057872, A083535:)
Digits:=2000;
tf:=[]; tr:=[]; tc:=[];
for n from 1 to 120 do
t2:=0;
j:=pi(n);
for i from 1 to j do t2:=t2+log(ithprime(i)); od;
tf:=[op(tf),floor(evalf(t2))];
tr:=[op(tr),round(evalf(t2))];
tc:=[op(tc),ceil(evalf(t2))];
od:

a(n) ~ n by the prime number theorem. - Charles R Greathouse IV, Aug 02 2012

A215260 Nearest integer to log(A002110(n)).

Original entry on oeis.org

0, 1, 2, 3, 5, 8, 10, 13, 16, 19, 23, 26, 30, 33, 37, 41, 45, 49, 53, 57, 62, 66, 70, 75, 79, 84, 88, 93, 98, 102, 107, 112, 117, 122, 127, 132, 137, 142, 147, 152, 157, 162, 167, 173, 178, 183, 189, 194, 199, 205, 210, 216, 221, 227, 232, 238, 243, 249, 254, 260, 266, 271, 277, 283, 288, 294, 300, 306, 312, 317, 323, 329, 335, 341, 347, 353, 359, 365, 371, 377

Offset: 0

N. J. A. Sloane, Sep 14 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A002110, A016040, A057872, A092777, A215259.

Round@ Log@ FoldList[ Times, 1, Prime@ Range@ 60] (* Robert G. Wilson v, Sep 16 2015 *)

v=List([t=0]);forprime(p=2,200,t+=log(p);listput(v,round(t))); Vec(v) \\ Charles R Greathouse IV, Sep 23 2012

a(n) = round(log(factorback(primes(n)))); \\ Michel Marcus, Jan 15 2025

a(n) ~ n log n, a statement equivalent to the Prime Number Theorem. - Charles R Greathouse IV, Sep 23 2012

A215259 Nearest integer to log(A003418(n)).

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 4, 6, 7, 8, 8, 10, 10, 13, 13, 13, 13, 16, 16, 19, 19, 19, 19, 22, 22, 24, 24, 25, 25, 28, 28, 32, 33, 33, 33, 33, 33, 36, 36, 36, 36, 40, 40, 44, 44, 44, 44, 48, 48, 49, 49, 49, 49, 53, 53, 53, 53, 53, 53, 58, 58, 62, 62, 62, 62, 62, 62, 67, 67, 67, 67, 71, 71, 75, 75, 75, 75, 75, 75, 79, 79, 81, 81, 85, 85, 85, 85, 85, 85, 89, 89, 89, 89, 89, 89

Offset: 0

N. J. A. Sloane, Sep 14 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

Cf. A003418, A215260, A057872.

v=List([0]);t=1;for(n=1,100,listput(v,round(log(t=lcm(t,n))))); Vec(v) \\ Charles R Greathouse IV, Sep 23 2012

a(n) ~ n, a statement equivalent to the Prime Number Theorem. - Charles R Greathouse IV, Sep 23 2012

A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 43, 47, 73, 103, 107, 109, 113, 199, 283, 467, 661, 887, 1063, 1069, 1097, 1103, 1109, 1123, 1129, 1303, 1307, 1321, 1327, 1621, 1627, 2803, 3931, 3947, 4273, 4289, 4297, 5867, 5869, 5881, 6373, 6379, 9439, 9473, 9479, 9497, 9551, 9859, 9931, 9949

Offset: 1

Jean-François Alcover, Dec 17 2014

nonn

The first 10000 terms are the same as A108310 (see that sequence for comments). - Charles R Greathouse IV, Dec 18 2014

This sequence, unlike A108310, is presumably infinite; it is finite if and only if theta(n) = n for some number n.

			Given that 1 - theta(3)/3 = 1 - log(6)/3 = 0.40..., 1 - theta(4)/4 = 1 - log(6)/4 = 0.55... and 1 - theta(5)/5 = 1 - log(30)/5 = 0.31..., the next term after 3 is 5.

Jean-François Alcover and Charles R Greathouse IV, Table of n, a(n) for n = 1..5000 (first 88 terms from Alcover)
Eric Weisstein's MathWorld, Chebyshev functions
Wikipedia, Chebyshev function

Cf. A016040, A035158, A057872, A083535, A108310, A215013.

(* Adapted from PARI *) Reap[For[record = 2; theta = 0; p = 2, p < 2 * 10^8, p = NextPrime[p], theta = theta + Log[p] //N; d = Abs[1 - theta/p]; If[d < record, record = d; Print[p]; Sow[p]]]][[2, 1]]

/* Note: This program may fail if you replace 1e6 with a number larger than 1e17, and will certainly fail at some point below 1e316. These large numbers are not remotely feasible at the moment. */
r=th=0; forprime(p=2,1e6, th+=log(p); t=th/p; if(t>r, r=t; print1(p", "); if(t>1, warning("theta(n) > n, possible missed terms")))) \\ Charles R Greathouse IV, Dec 17 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

SageMath

Formula

Extensions

A035158 Floor of the Chebyshev function theta(n): a(n) = floor(Sum_{primes p <= n } log(p)).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Formula

A215260 Nearest integer to log(A002110(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A215259 Nearest integer to log(A003418(n)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A252398 Successive n with minimal relative distance |1-theta(n)/n|, where theta(n) = log(A034386(n)) is Chebyshev's theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI