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, Mathematics 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)/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
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
KEYWORD
nonn,frac
AUTHOR
STATUS
approved