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

1,2

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.

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)

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

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

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

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

Cf. A236435, A236436.

Cf. A001511, A072078, A355583.

Sequence in context: A244377 A244330 A342867 * A241198 A356094 A296296

Adjacent sequences: A060750 A060751 A060752 * A060754 A060755 A060756

Keyword

nonn,frac

Author

Frank Ellermann, Apr 23 2001

Extensions

Definition corrected by Jonathan Sondow, Jan 31 2014

Status

approved