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 A038110 Numerator of frequency of integers with smallest divisor prime(n). 32
 1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Numerator of Product_{k=1..n-1} (1 - 1/prime(k)). - Jonathan Sondow, Jan 31 2014 Equivalently, denominator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A060753). - N. J. A. Sloane, Apr 17 2015 Sum_{n>=1} a(n)/A038111(n) = 1. - Bob Selcoe, Jan 09 2015 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 LINKS Robert Israel, Table of n, a(n) for n = 1..278 F. Ellermann, Illustration for A002110, A005867, A038110, A060753 Fred Kline and Gerry Myerson, Identity for frequency of integers with smallest prime(n) divisor, Math Stack Exchange question V. 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). J. Sondow and Eric Weisstein, Euler Product, World of Mathematics Wikipedia, Mertens' theorems FORMULA a(n) = A005867(n) / gcd( A005867(n), A002110(n) ). 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 a(n+1)/A038111(n+1) = a(n)/A038111(n) * (prime(n)-1)/prime(n+1). - Robert Israel, Jul 14 2014 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 EXAMPLE a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ). MAPLE N:= 100: # for a(1) to a(N) Q:= 1: p:= 1: for n from 1 to N do   p:= nextprime(p);   A[n]:= numer(Q);   Q:= Q * (1 - 1/p); end: seq(A[n], n=1..N); # Robert Israel, Jul 14 2014 MATHEMATICA Numerator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 64} ] (* Wouter Meeussen *) Numerator@Table[ Product[ 1 - 1/Prime[ k ], {k, n-1}], {n, 64} ] (* Jonathan Sondow, Jan 31 2014 *) Numerator@ Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/ Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 21}] (* Fred Daniel Kline, Jul 14 2014 *) PROG (PARI) a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ Michel Marcus, Aug 05 2019 CROSSREFS Cf. A038111, A002110, A005867, A060753, A236435, A236436, A254196. Sequence in context: A230112 A023376 A242966 * A241197 A130436 A260306 Adjacent sequences:  A038107 A038108 A038109 * A038111 A038112 A038113 KEYWORD nonn,frac AUTHOR STATUS approved

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Last modified May 29 06:03 EDT 2020. Contains 334697 sequences. (Running on oeis4.)