A038111 Denominator of density of integers with smallest prime factor prime(n).

2, 6, 15, 105, 385, 1001, 17017, 323323, 7436429, 19605131, 86822723, 3212440751, 131710070791, 5663533044013, 266186053068611, 613385252723321, 2783825377744303, 5855632691117327, 392327390304860909, 27855244711645124539, 2033432863950094091347, 160641196252057433216413

Offset: 1

Author

Wouter Meeussen

Keywords

• nonn
• frac

Comments

Denominator of (Product_{k=1..n-1} (1 - 1/prime(k)))/prime(n). - Vladimir Shevelev, Jan 09 2015

a(n)/a(n-1) = prime(n)/q(n) where q(n) is 1 or a prime for all n < 1000. What are the first indices for which q(n) is composite? - M. F. Hasler, Dec 04 2018

Examples

			From _M. F. Hasler_, Dec 03 2018: (Start)
The density of the even numbers is 1/2, thus a(1) = 2.
The density of the numbers divisible by 3 but not by 2 is 1/6, thus a(2) = 6.
The density of multiples of 5 not divisible by 2 or 3 is 2/30, thus a(3) = 15. (End)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..277
• Fred Kline and Gerry Myerson, Identity for frequency of integers with smallest prime(n) divisor, Mathematics Stack Exchange, Jul 2014.
• Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int'l J. Math. and Math. Sci. (2008) Art. ID 908045, 1-12. See Eq. (5.8).

Crossrefs

Cf. A038110, A060753, A112037.

Programs

Maple

N:= 100: # for the first N terms
Q:= 1: p:= 1:
for n from 1 to N do
  p:= nextprime(p);
  A[n]:= denom(Q/p);
  Q:= Q * (1 - 1/p);
end:
seq(A[n],n=1..N); # Robert Israel, Jul 14 2014

Mathematica

Denominator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 1, 64} ]
(* Wouter Meeussen *)
Denominator@
Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/
Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 1, 21}]
(* Fred Daniel Kline, Jul 14 2014 *)

PARI

apply( A038111(n)=denominator(prod(k=1,n-1,1-1/prime(k)))*prime(n), [1..30]) \\ M. F. Hasler, Dec 03 2018

Formula

a(n) = denominator of phi(e^(psi(p_n-1)))/e^(psi(p_n)), where psi(.) is the second Chebyshev function and phi(.) is Euler's totient function. - Fred Daniel Kline, Jul 17 2014

a(n) = prime(n)*A060753(n). - Vladimir Shevelev, Jan 10 2015

a(n) = a(n-1)*prime(n)/q(n), where q(n) = 1 except for q({3, 5, 6, 10, 11, 16, 17, 18, ...}) = (2, 3, 5, 11, 7, 23, 13, 29, ...), cf. A112037. - M. F. Hasler, Dec 03 2018

Extensions

Name edited by M. F. Hasler, Dec 03 2018