A236435 -id:A236435 - cp's OEIS frontend

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

Wouter Meeussen

Numerator of Product_{k=1..n-1} (1 - 1/prime(k)). - Jonathan Sondow, Jan 31 2014
Equivalently, denominator of Product_{k=1..n-1} prime(k)/(prime(k)-1) (cf. A060753). - N. J. A. Sloane, Apr 17 2015
Sum_{n>=1} a(n)/A038111(n) = 1. - Bob Selcoe, Jan 09 2015
a(n)/A038111(n) = (1/prime(n))*Product_{k=1..n-1} (1 - 1/prime(k)) ~ e^(-c)/ (prime(n)*log(prime(n))), where c=0.577... is the Euler constant. - Vladimir Shevelev, Jan 10 2015

			a(10) = 110592 = ( 1*2*4*6*10*12*16*18*22 ) / ( 2*3*5*11 ).

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

Cf. A038111, A002110, A005867, A058250, A060753, A236435, A236436, A254196.

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

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 *)

a(n) = numerator(prod(k=1, n-1, (1 - 1/prime(k)))); \\ Michel Marcus, Aug 05 2019

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+1)/A038111(n+1) = a(n)/A038111(n) * (prime(n)-1)/prime(n+1). - Robert Israel, Jul 14 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

A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).

Original entry on oeis.org

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

Offset: 1

Frank Ellermann, Apr 23 2001

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)

			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

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

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.

Cf. A000203, A002110, A005867, A038110, A038111.
Cf. A236435, A236436.
Cf. A001511, A072078, A355583.

[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

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 *)

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

Definition corrected by Jonathan Sondow, Jan 31 2014

A236436 Denominator of product_{k=1..n-1} (1 + 1/prime(k)).

Original entry on oeis.org

1, 2, 1, 5, 35, 385, 715, 12155, 46189, 1062347, 30808063, 955049953, 1859834119, 76253198879, 298080686527, 14009792266769, 742518990138757, 43808620418186663, 86204059532560853, 339745411098916303, 24121924188023057513, 47591904479072518877, 3759760453846728991283

Offset: 1

Jonathan Sondow, Feb 01 2014

			(1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has denominator a(5) = 35.

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. Sondow and E. Weisstein, MathWorld: Mertens Theorem

Cf. A038110, A060753, A072044, A072045, A236435.

Denominator@Table[Product[1 + 1/Prime[k], {k, 1, n - 1}], {n, 1, 23}]

A236435(n+1) / a(n+1) = A072045(n)/A072044(n) / A038110(n+1)/A060753(n+1) because 1+x = (1-x^2) / (1-x).

A236436(n) / a(n) = product_{k=1..n-1} (1 + 1/prime(k)) ~ (6/Pi^2)*exp(gamma)*log(n) as n -> infinity, by Mertens' theorem.

A072044 Numerator of Product{prime(k)^2/(prime(k)^2 - 1) | 0A072045.

Original entry on oeis.org

1, 4, 3, 25, 1225, 29645, 715715, 206841635, 14933966047, 718188003533, 86285158710179, 82920037520482019, 5974606913975783369, 10043314222393291843289, 1688189817927745147112851, 162139622078364740433577733, 35034630647548196605993834769

Offset: 0

Reinhard Zumkeller, Jun 09 2002

a(n)/A072045(n) -> (Pi^2)/6 (Leonhard Euler, 1748).

			For the first 3 primes: 2,3,5: (2^2/(2^2-1))*(3^2/(3^2-1))*(5^2/(5^2-1)) = (4/3)*(9/8)*(25/24) = (4*9*25)/(3*8*24) = 25/16, therefore a(3)=25;
a(10)/A072045(10)=86285158710179/52836150804480=1.63307049.

M. Sigg: "Pi" p. 191 in Lexikon der Mathematik, Band 4, Spektrum Verlag, 2002.

Cf. A000040, A001248, A236435, A236436.

Numerator/@Rest[FoldList[Times,1,#/(#-1)&/@(Prime[Range[15]]^2)]] (* Harvey P. Dale, May 03 2011 *)

a(0)=1 prepended by Alois P. Heinz, May 11 2024

A072045 Denominator of Product{prime(k)^2/(prime(k)^2 - 1) | 0A072044.

Original entry on oeis.org

1, 3, 2, 16, 768, 18432, 442368, 127401984, 9172942848, 440301256704, 52836150804480, 50722704772300800, 3652034743605657600, 6135418369257504768000, 1030750286035260801024000, 98952027459385036898304000, 21373637931227167970033664000

Offset: 0

Reinhard Zumkeller, Jun 09 2002

A072044(n)/a(n) -> (Pi^2)/6 (Leonhard Euler, 1748).

			For the first 3 primes: 2,3,5: (2^2/(2^2-1))*(3^2/(3^2-1))*(5^2/(5^2-1)) = (4/3)*(9/8)*(25/24) = (4*9*25)/(3*8*24) = 25/16, therefore a(3)=16;
A072044(9)/a(9)=718188003533/440301256704=1.631128671.

M. Sigg: "Pi" p. 191 in Lexikon der Mathematik, Band 4, Spektrum Verlag, 2002.

Cf. A000040, A001248, A236435, A236436.

Rest[Denominator[FoldList[Times,1,(#^2/(#^2-1)&/@Prime[Range[20]])]]] (* Harvey P. Dale, Oct 14 2012 *)

More terms from Harvey P. Dale, Oct 14 2012

a(0)=1 prepended by Alois P. Heinz, May 11 2024

A335004 Decimal expansion of 6*exp(gamma)/Pi^2.

Original entry on oeis.org

1, 0, 8, 2, 7, 6, 2, 1, 9, 3, 2, 6, 0, 9, 2, 4, 5, 8, 0, 1, 2, 2, 1, 8, 8, 0, 3, 8, 1, 9, 0, 9, 2, 6, 5, 7, 0, 1, 8, 4, 3, 0, 6, 6, 5, 5, 5, 8, 3, 6, 0, 0, 1, 4, 4, 1, 0, 2, 0, 3, 1, 9, 7, 4, 3, 5, 5, 1, 2, 8, 6, 1, 9, 2, 9, 8, 2, 9, 5, 0, 4, 3, 4, 2, 4, 2, 2

Offset: 1

Amiram Eldar, May 19 2020

			1.0827621932609245801221880381909265701843066555836...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.1, p. 31.
József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, page 100.

J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Jean M. de Koninck and Claude Levesque (eds.), Théorie des nombres/Number theory: Proceedings of the International Number Theory Conference held at Université Laval, July 5-18, 1987, Berlin, New York: de Gruyter, 1989, pp. 201-206.
Florian Luca and Carl Pomerance, On some problems of Mąkowski-Schinzel and Erdős concerning the arithmetical functions phi and sigma, Colloquium Mathematicum, Vol. 92 (2002), pp. 111-130.
Michel Planat, Riemann hypothesis from the Dedekind psi function, arXiv:1010.3239 [math.GM], 2010.

Cf. A001620 (gamma), A013661 (Pi^2/6), A051377 (esigma), A059956 (6/Pi^2), A073004 (exp(gamma)), A246499 (Pi^2/(6*exp(gamma))).

Cf. A236435, A236436.

RealDigits[6*Exp[EulerGamma]/Pi^2, 10, 100][[1]]

6*exp(Euler)/Pi^2 \\ Michel Marcus, May 19 2020

Equals limsup_{k->oo} esigma(k)/(k*log(log(k))), where esigma(k) is the sum of exponential divisors of k (A051377).
Equals A073004 * A059956 = A073004 / A013661 = 1 / A246499.
Equals lim_{k->oo} (1/log(k)) * Product_{p prime <= k} (1 + 1/p). - Amiram Eldar, Jul 09 2020

A382489 The number of unitary 5-smooth divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8, 1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Mar 29 2025

Period 30: repeat [1, 2, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 4, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 2, 1, 8].
In general, the sequence of the number of unitary prime(k)-smooth divisors of n, for k >= 1, is periodic with period A002110(k).
Decimal expansion of 135804580460138015713571358020/111111111111111111111111111111.
Continued fraction expansion of 808690/(525316 + sqrt(382161348866)) (with offset 0).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A002110, A005117, A034444, A051037, A054640, A236435, A236436, A355582, A355583.

The number of unitary prime(k)-smooth divisors of n: A134451 (k = 1), A382488 (k = 2), this sequence (k = 3).

a[n_] := Product[If[Divisible[n, p], 2, 1], {p, {2, 3, 5}}]; Array[a, 100]

a(n) = vecprod(apply(x -> !((n % 30) % x) + 1, [2, 3, 5]))

Multiplicative with a(p^e) = 2 if p <= 5, and 1 otherwise.
a(n) = A034444(A355582(n)).
a(n) = A034444(n) if and only if n is 5-smooth (A051037).
a(n) = A355583(n) if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 12/5.
In general, the asymptotic mean of the number of unitary prime(k)-smooth divisors of n is A054640(k)/A002110(k) = A236435(k)/A236436(k).
Dirichlet g.f.: (1 + 1/2^s) * (1 + 1/3^s) * (1 + 1/5^s) * zeta(s).
In general, Dirichlet g.f. of the number of unitary prime(k)-smooth divisors of n is zeta(s) * Product_{p prime <= prime(k)} (1 + 1/p^s).

cp's OEIS Frontend

A038110 Numerator of frequency of integers with smallest divisor prime(n).

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

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A236436 Denominator of product_{k=1..n-1} (1 + 1/prime(k)).

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A072044 Numerator of Product{prime(k)^2/(prime(k)^2 - 1) | 0A072045.

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A072045 Denominator of Product{prime(k)^2/(prime(k)^2 - 1) | 0A072044.

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A335004 Decimal expansion of 6*exp(gamma)/Pi^2.

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A382489 The number of unitary 5-smooth divisors of n.

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A060753 Denominator of 1246...(prime(n-1)-1) / (2357...prime(n-1)).