OFFSET
1,2
COMMENTS
Equivalently, numerator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A038110). - N. J. A. Sloane, Apr 17 2015
a(n)/A038110(n) is the supremum of the abundancy index sigma(k)/k = A000203(k)/k of the prime(n-1)-smooth numbers, for n>1 (Laatsch, 1986). - Amiram Eldar, Oct 26 2021
From Amiram Eldar, Jul 10 2022: (Start)
a(n)/A038110(n) is the sum of the reciprocals of the prime(n-1)-smooth numbers, for n>1.
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 429.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..423
Frank Ellermann, Illustration for A002110, A005867, A038110, A060753.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92.
Jonathan Sondow and Eric Weisstein, Euler Product, MathWorld.
FORMULA
A038110(n) / a(n) ~ exp( -gamma ) / log( prime(n) ), Mertens's theorem for x = prime(n) = A000040(n).
a(n) = A038111(n) / prime(n). - Vladimir Shevelev, Jan 10 2014
EXAMPLE
A038110(50)/ a(50) = 0.1020..., exp(-gamma)/log(229) = 0.1033...
1*2*4/(2*3*5) = 4/15 has denominator a(4) = 15. - Jonathan Sondow, Jan 31 2014
MATHEMATICA
Table[Denominator@ Product[EulerPhi@ Prime[i]/Prime@ i, {i, n}], {n, 0, 19}] (* Michael De Vlieger, Jan 10 2015 *)
{1}~Join~Denominator@ FoldList[Times, Table[EulerPhi@ Prime[n]/Prime@ n, {n, 19}]] (* Michael De Vlieger, Jul 26 2016 *)
b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]
Denominator@ Table[b[n], {n, 0, 20}] (* Fred Daniel Kline, Jun 27 2017 *)
Join[{1}, Denominator[With[{nn=20}, FoldList[Times, Prime[Range[nn]]-1]/FoldList[ Times, Prime[Range[nn]]]]]] (* Harvey P. Dale, Apr 17 2022 *)
PROG
(Magma) [1] cat [Denominator((&*[NthPrime(k-1)-1:k in [2..n]])/(&*[NthPrime(k-1):k in [2..n]])):n in [2..20]]; // Marius A. Burtea, Sep 19 2019
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Frank Ellermann, Apr 23 2001
EXTENSIONS
Definition corrected by Jonathan Sondow, Jan 31 2014
STATUS
approved