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A038110
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Numerator of frequency of integers with smallest divisor prime(n).
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40
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1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000
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OFFSET
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1,4
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COMMENTS
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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
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
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LINKS
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J. Sondow and Eric Weisstein, Euler Product, World of Mathematics
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FORMULA
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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) = 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
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EXAMPLE
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a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ).
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MAPLE
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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:
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MATHEMATICA
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Numerator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 64} ]
Numerator@Table[ Product[ 1 - 1/Prime[ k ], {k, n-1}], {n, 64} ]
Numerator@
Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/
Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 21}]
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PROG
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(PARI) a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ Michel Marcus, Aug 05 2019
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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