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A038110 Numerator of frequency of integers with smallest divisor prime(n). 34

%I

%S 1,1,1,4,8,16,192,3072,55296,110592,442368,13271040,477757440,

%T 19110297600,802632499200,1605264998400,6421059993600,12842119987200,

%U 770527199232000,50854795149312000,3559835660451840000

%N Numerator of frequency of integers with smallest divisor prime(n).

%C Numerator of Product_{k=1..n-1} (1 - 1/prime(k)). - _Jonathan Sondow_, Jan 31 2014

%C Equivalently, denominator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A060753). - _N. J. A. Sloane_, Apr 17 2015

%C Sum_{n>=1} a(n)/A038111(n) = 1. - _Bob Selcoe_, Jan 09 2015

%C a(n)/A038111(n) = (1/prime(n))*Product_{k=1..n-1} (1 - 1/prime(k)) ~ e^(-c)/ (prime(n)*log(prime(n))), where c=0.577... is the Euler constant. - _Vladimir Shevelev_, Jan 10 2015

%H Robert Israel, <a href="/A038110/b038110.txt">Table of n, a(n) for n = 1..278</a>

%H F. Ellermann, <a href="/A005867/a005867.txt">Illustration for A002110, A005867, A038110, A060753</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>, Math Stack Exchange question

%H V. 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).

%H J. Sondow and Eric Weisstein, <a href="http://mathworld.wolfram.com/EulerProduct.html">Euler Product</a>, World of Mathematics

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mertens%27_theorems">Mertens' theorems</a>

%F a(n) = A005867(n) / gcd( A005867(n), A002110(n) ).

%F a(n)/A060753(n) = Product_{k=1..n-1} (1 - 1/prime(k)) ~ exp(-gamma)/log(n) as n->infinity (Mertens's 3rd theorem). - _Jonathan Sondow_, Jan 31 2014

%F a(n+1)/A038111(n+1) = a(n)/A038111(n) * (prime(n)-1)/prime(n+1). - _Robert Israel_, Jul 14 2014

%F a(n) = numerator 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

%e a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ).

%p N:= 100: # for a(1) to a(N)

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

%p for n from 1 to N do

%p p:= nextprime(p);

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

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

%p end:

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

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

%t (* _Wouter Meeussen_ *)

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

%t (* _Jonathan Sondow_, Jan 31 2014 *)

%t Numerator@

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

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

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

%o (PARI) a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ _Michel Marcus_, Aug 05 2019

%Y Cf. A038111, A002110, A005867, A060753, A236435, A236436, A254196.

%K nonn,frac

%O 1,4

%A _Wouter Meeussen_

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Last modified July 4 12:18 EDT 2020. Contains 335448 sequences. (Running on oeis4.)