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

1,1

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

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, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12. Eq. (5.8).

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

Example

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)

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 *)

Prog

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

Crossrefs

Cf. A038110, A060753, A112037.

Sequence in context: A007709 A190339 A078328 * A356803 A261726 A302775

Adjacent sequences: A038108 A038109 A038110 * A038112 A038113 A038114

Keyword

nonn,frac

Author

Wouter Meeussen

Extensions

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

Status

approved