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
Examples
a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ).
Links
- Robert Israel, Table of n, a(n) for n = 1..278
- Frank 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
- Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int'l J. Math. and Math. Sci. (2008) Art. ID 908045, 1-12. See Eq. (5.8).
- Jonathan Sondow and Eric Weisstein, Euler Product, World of Mathematics
- Wikipedia, Mertens' theorems
Programs
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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
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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 *) -
PARI
a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ Michel Marcus, Aug 05 2019
Formula
a(n) = A005867(n-1) / A058250(n-1), where A058250(m) = gcd(A005867(m), A002110(m)). [Edited by Peter Munn, Jun 29 2025]
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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