A038111 - OEIS

%I #71 Aug 12 2022 19:46:09

%S 2,6,15,105,385,1001,17017,323323,7436429,19605131,86822723,

%T 3212440751,131710070791,5663533044013,266186053068611,

%U 613385252723321,2783825377744303,5855632691117327,392327390304860909,27855244711645124539,2033432863950094091347,160641196252057433216413

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

%C Denominator of (Product_{k=1..n-1} (1 - 1/prime(k)))/prime(n). - _Vladimir Shevelev_, Jan 09 2015

%C 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

%H Robert Israel, <a href="/A038111/b038111.txt">Table of n, a(n) for n = 1..277</a>

%H Fred Kline and Gerry Myerson, <a href="http://math.stackexchange.com/q/867135/28555">Identity for frequency of integers with smallest prime(n) divisor</a>, Mathematics Stack Exchange, Jul 2014.

%H Vladimir Shevelev, <a href="http://www.hindawi.com/journals/ijmms/2008/908045.html">Generalized Newman phenomena and digit conjectures on primes</a>, Internat. J. of Mathematics and Math. Sciences, 2008 (2008), Article ID 908045, 1-12. Eq. (5.8).

%F 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

%F a(n) = prime(n)*A060753(n). - _Vladimir Shevelev_, Jan 10 2015

%F 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

%e From _M. F. Hasler_, Dec 03 2018: (Start)

%e The density of the even numbers is 1/2, thus a(1) = 2.

%e The density of the numbers divisible by 3 but not by 2 is 1/6, thus a(2) = 6.

%e The density of multiples of 5 not divisible by 2 or 3 is 2/30, thus a(3) = 15. (End)

%p N:= 100: # for the first N terms

%p Q:= 1: p:= 1:

%p for n from 1 to N do

%p   p:= nextprime(p);

%p   A[n]:= denom(Q/p);

%p   Q:= Q * (1 - 1/p);

%p end:

%p seq(A[n],n=1..N); # _Robert Israel_, Jul 14 2014

%t Denominator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 1, 64} ]

%t (* _Wouter Meeussen_ *)

%t Denominator@

%t Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/

%t Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 1, 21}]

%t (* _Fred Daniel Kline_, Jul 14 2014 *)

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

%Y Cf. A038110, A060753, A112037.

%K nonn,frac

%O 1,1

%A _Wouter Meeussen_

%E Name edited by _M. F. Hasler_, Dec 03 2018