A072044 -id:A072044 - cp's OEIS frontend

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

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

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.

A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

Original entry on oeis.org

1, 3, 2, 12, 96, 1152, 2304, 41472, 165888, 3981312, 119439360, 3822059520, 7644119040, 321052999680, 1284211998720, 61642175938560, 3328677500682240, 199720650040934400, 399441300081868800, 1597765200327475200, 115039094423578214400, 230078188847156428800, 18406255107772514304000

Offset: 1

Jonathan Sondow, Feb 01 2014

A236436(n)/(a(n)*zeta(2)) is the asymptotic density of the prime(n-1)-rough squarefree numbers (squarefree numbers whose prime factors are all >= prime(n-1)) for n >= 2. E.g., A236436(2)/(a(2)*zeta(2)) = 2/(3*zeta(2)) = 4/Pi^2 (A185199) is the asymptotic density of the odd squarefree numbers (A056911), and A236436(3)/(a(3)*zeta(2)) = 1/(2*zeta(2)) = 3/Pi^2 (A104141) is the asymptotic density of the 5-rough squarefree numbers (A276378). - Amiram Eldar, Aug 26 2025

			(1 + 1/2)*(1 + 1/3)*(1 + 1/5)*(1 + 1/7) = 96/35 has numerator a(5) = 96.
Fractions begin with 1, 3/2, 2, 12/5, 96/35, 1152/385, 2304/715, 41472/12155, 165888/46189, 3981312/1062347, 119439360/30808063, 3822059520/955049953, ...

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
Jonathan Sondow and Eric W. Weisstein, MathWorld: Mertens Theorem.

Cf. A038110, A060753, A072044, A072045, A073004, A236436 (denominators), A335004.

Cf. A013661, A056911, A104141, A185199, A276378.

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

a(n+1) / A236436(n+1) = (A072045(n)/A072044(n)) / (A038110(n+1)/A060753(n+1)) because 1+x = (1-x^2) / (1-x).

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

A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

Original entry on oeis.org

1, 3, 4, 7, 5, 12, 7, 15, 13, 15, 11, 28, 13, 21, 20, 31, 17, 39, 19, 35, 28, 33, 23, 60, 25, 39, 40, 49, 29, 60, 31, 63, 44, 51, 35, 91, 37, 57, 52, 75, 41, 84, 43, 77, 65, 69, 47, 124, 49, 75, 68, 91, 53, 120, 55, 105, 76, 87, 59, 140, 61, 93, 91, 127, 65, 132

Offset: 1

Amiram Eldar, Jun 19 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586, A064987, A072044, A072045, A072079.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), this sequence (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p < 5, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p < 5, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385138(n).
Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= 3, and p^e if p >= 5.
In general, the sum of divisors d of n such that n/d is q-smooth (not divisible by a prime larger than q) is multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if p <= q, and p^e if p > q.
Dirichlet g.f.: zeta(s-1) / ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-smooth has Dirichlet g.f.: zeta(s-1) / Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (3/4)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-smooth has an average order c * n^2 / 2, where c = A072044(k-1)/A072045(k-1) for k >= 2.

A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

Original entry on oeis.org

1, 2, 3, 4, 6, 6, 8, 8, 9, 12, 12, 12, 14, 16, 18, 16, 18, 18, 20, 24, 24, 24, 24, 24, 31, 28, 27, 32, 30, 36, 32, 32, 36, 36, 48, 36, 38, 40, 42, 48, 42, 48, 44, 48, 54, 48, 48, 48, 57, 62, 54, 56, 54, 54, 72, 64, 60, 60, 60, 72, 62, 64, 72, 64, 84, 72, 68, 72

Offset: 1

Amiram Eldar, Jun 19 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007310, A013661, A064987, A072044, A072045, A186099.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), A385134 (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), this sequence (5-rough), A385139 (exponentially 2^n).

f[p_, e_] := If[p > 3, (p^(e+1) - 1)/(p - 1), p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(p > 3, (p^(e + 1) - 1)/(p - 1), p^e));}

a(n) = A064987(n)/A385137(n).
Multiplicative with a(p^e) = p^e if p <= 3, and (p^(e+1)-1)/(p-1) if p >= 5.
In general, the sum of divisors d of n such that n/d is q-rough (not divisible by a prime smaller than q) is multiplicative with a(p^e) = p^e if p <= q, and (p^(e+1)-1)/(p-1) if p > q.
Dirichlet g.f.: zeta(s-1) * zeta(s) * ((1 - 1/2^s) * (1 - 1/3^s)).
In general, the sum of divisors d of n such that n/d is q-rough has Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime <= q} (1 - 1/q^s).
Sum_{k=1..n} a(k) ~ (Pi^2/18)*n^2.
In general, the sum of divisors d of n such that n/d is prime(k)-rough has an average order c * n^2 / 2, where c = zeta(2) * A072045(k-1)/A072044(k-1) for k >= 2.

A372634 Numerators of the reduced fraction of all rational number representations (a/b, with a and b being integers) which can themselves be reduced by at least one common prime factor of at most prime(n).

Original entry on oeis.org

1, 1, 9, 457, 11213, 273347, 79439651, 5761023199, 277886746829, 33449007905699, 32197332748181219, 2322572170370125769, 3907895853135787075289, 657439531892484346088851, 63187594618979703535273733, 13660992716321028635960170769

Offset: 1

Brian Lee Burtner, May 08 2024

The numerators of the fraction F(n) = a(n)/A072044(n) may be generated directly by use of inclusion-exclusion; e.g., 1/4 + 1/9 + 1/25 - 1/225 - 1/100 - 1/36 + 1/900 = 9/25.
Following Euler, they may also be generated via products.
a(n)/A072044(n) -> 1 - 6/Pi^2 (provable via Euler, see references).  This value is the supremal proportion of all rational number representations a/b that are reducible by some common factor (or, more broadly: the proportion of all pairs of integers a,b that are not coprime).
A072045(n)/A072044(n) gives the complementary proportion of all rational number representations that are irreducible by any prime factor of at most A000040(n).  This analogously converges to 6/Pi^2, the infimal proportion of all rational number representations a/b that are simply irreducible.

			For n=3, 1 - (3/4)*(8/9)*(24/25) = 9/25.
Exactly 9/25 of all rational number representations are reducible by at least one prime factor of at most 5.

Leonhard Euler, Introductio In Analysin Infinitorum Vol 1, 1748, p. 474.

Leonhard Euler, Introductio in Analysin Infinitorum, Vol 1.
Wikipedia, Probability of coprimality.

Cf. A072044, A072045, A229099.

b:= proc(n) b(n):= `if`(n=0, 1, b(n-1)*(1-1/ithprime(n)^2)) end:
a:= n-> numer(1-b(n)):
seq(a(n), n=1..16);  # Alois P. Heinz, May 11 2024

a[n_]:=Numerator[1-Product[1-1/Prime[k]^2,{k,n}]]; Array[a,16] (* Stefano Spezia, May 11 2024 *)

a(n) = numerator(1 - prod(k=1, n, (prime(k)^2-1)/prime(k)^2)); \\ Michel Marcus, May 08 2024

a(n) = numerator(1 - Product_{k=1..n} (1 - 1/prime(k)^2)).

cp's OEIS Frontend

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

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

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A236435 Numerator of Product_{k=1..n-1} (1 + 1/prime(k)).

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A385137 The sum of divisors d of n such that n/d is a 3-smooth number (A003586).

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A385138 The sum of divisors d of n such that n/d is a 5-rough number (A007310).

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A372634 Numerators of the reduced fraction of all rational number representations (a/b, with a and b being integers) which can themselves be reduced by at least one common prime factor of at most prime(n).

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