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

1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000

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) = 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: A023376 A242966 A337235 * A241197 A130436 A260306

Adjacent sequences: A038107 A038108 A038109 * A038111 A038112 A038113

Keyword

nonn,frac

Author

Wouter Meeussen

Status

approved