A060753 Denominator of 1*2*4*6*...*(prime(n-1)-1) / (2*3*5*7*...*prime(n-1)).

1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909

Offset: 1

Author

Frank Ellermann, Apr 23 2001

Keywords

• nonn
• frac

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.

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)

Examples

			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
		

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.

Crossrefs

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

Cf. A236435, A236436.

Cf. A001511, A072078, A355583.

Programs

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

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 *)

Formula

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

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

A038110(n) / a(n) = A005867(n) / A002110(n). - corrected by Simon Tatham, Jul 26 2016

a(n) = A038111(n) / prime(n). - Vladimir Shevelev, Jan 10 2014

a(n) = A038110(n) + A161527(n-1). - Jamie Morken, Jun 19 2019

Extensions

Definition corrected by Jonathan Sondow, Jan 31 2014