A060753 - OEIS

%I #107 Jan 09 2023 07:11:41

%S 1,2,3,15,35,77,1001,17017,323323,676039,2800733,86822723,3212440751,

%T 131710070791,5663533044013,11573306655157,47183480978717,

%U 95993978542907,5855632691117327,392327390304860909

%N Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).

%C Equivalently, numerator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A038110). - _N. J. A. Sloane_, Apr 17 2015

%C 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

%C From _Amiram Eldar_, Jul 10 2022: (Start)

%C a(n)/A038110(n) is the sum of the reciprocals of the prime(n-1)-smooth numbers, for n>1.

%C a(n)/A038110(n) is the asymptotic mean of the number of prime(n-1)-smooth divisors of the positive integers, for n>1 (cf. A001511, A072078, A355583). (End)

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 429.

%H Michael De Vlieger, <a href="/A060753/b060753.txt">Table of n, a(n) for n = 1..423</a>

%H Frank Ellermann, <a href="/A005867/a005867.txt">Illustration for A002110, A005867, A038110, A060753</a>.

%H Richard Laatsch, <a href="http://www.jstor.org/stable/2690424">Measuring the abundancy of integers</a>, Mathematics Magazine, Vol. 59, No. 2 (1986), pp. 84-92.

%H Jonathan Sondow and Eric Weisstein, <a href="http://mathworld.wolfram.com/EulerProduct.html">Euler Product</a>, MathWorld.

%F a(n) = A002110(n) / gcd( A005867(n), A002110(n) ).

%F A038110(n) / a(n) ~ exp( -gamma ) / log( prime(n) ), Mertens's theorem for x = prime(n) = A000040(n).

%F A038110(n) / a(n) = A005867(n) / A002110(n). - corrected by _Simon Tatham_, Jul 26 2016

%F a(n) = A038111(n) / prime(n). - _Vladimir Shevelev_, Jan 10 2014

%F a(n) = A038110(n) + A161527(n-1). - _Jamie Morken_, Jun 19 2019

%e A038110(50)/ a(50) = 0.1020..., exp(-gamma)/log(229) = 0.1033...

%e 1*2*4/(2*3*5) = 4/15 has denominator a(4) = 15. - _Jonathan Sondow_, Jan 31 2014

%t Table[Denominator@ Product[EulerPhi@ Prime[i]/Prime@ i, {i, n}], {n, 0, 19}] (* _Michael De Vlieger_, Jan 10 2015 *)

%t {1}~Join~Denominator@ FoldList[Times, Table[EulerPhi@ Prime[n]/Prime@ n, {n, 19}]] (* _Michael De Vlieger_, Jul 26 2016 *)

%t b[0] := 0; b[n_] := b[n - 1] + (1 - b[n - 1]) / Prime[n]

%t Denominator@ Table[b[n], {n, 0, 20}] (* _Fred Daniel Kline_, Jun 27 2017 *)

%t Join[{1},Denominator[With[{nn=20},FoldList[Times,Prime[Range[nn]]-1]/FoldList[ Times,Prime[Range[nn]]]]]] (* _Harvey P. Dale_, Apr 17 2022 *)

%o (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

%Y Cf. A000203, A002110, A005867, A038110, A038111.

%Y Cf. A236435, A236436.

%Y Cf. A001511, A072078, A355583.

%K nonn,frac

%O 1,2

%A _Frank Ellermann_, Apr 23 2001

%E Definition corrected by _Jonathan Sondow_, Jan 31 2014