A335004 -id:A335004 - cp's OEIS frontend

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

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.

A369319 a(n) is the sum of the greatest common exponential divisor of n and k over the positive numbers k that do not exceed n.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 7, 12, 12, 10, 11, 18, 13, 14, 15, 24, 17, 24, 19, 30, 21, 22, 23, 36, 30, 26, 33, 42, 29, 30, 31, 40, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 72, 56, 60, 51, 78, 53, 66, 55, 84, 57, 58, 59, 90, 61, 62, 84, 84, 65, 66, 67

Offset: 1

Amiram Eldar, Feb 13 2024

First differs from A336465 at n = 27.
The sum is restricted to numbers k that have a common exponential divisor with n, i.e., numbers k with rad(k) = rad(n), where rad is the squarefree kernel function (A007947).
Analogous to Pillai's arithmetical function (A018804), with exponential divisors instead of divisors.

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; arXiv preprint, arXiv:math/0610274v2 [math.NT], 2006-2009.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000010, A007947, A018804, A072911, A308056, A335004, A336465.

f[p_, e_] := DivisorSum[e, p^#*EulerPhi[e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, f[i,1]^d * eulerphi(f[i,2]/d)));}

a(n) = Sum_{k=1..n, rad(k) = rad(n)} (n, k)(e), where (n, k)(e) = Product_{p|n} p^gcd(v_p(n), v_p(k)), and v_p(n) is the p-adic valuation of n (the exponent of the highest power of p that divides n).
Multiplicative with a(p^e) = Sum_{k=1..e} p^gcd(e, k) = Sum_{d|e} p^d * phi(e/d), where phi is the Euler totient function (A000010).
Dirichlet g.f.: (zeta(s-1)*zeta(2*s-1)/zeta(3*s-2)) * Product_{p prime} (1 + ((p^(s-1)-1)*(p^(2*s-1)-1)/(p^(3*s-2)-1)) * Sum_{k>=3} phi(k)/(p^(k*s-1)-1)).
Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n * log(n)^(5/3)), where c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 1.16509457249412700814... .
Lim sup_{n->oo} a(n)/(n*log(log(n))) = 6 * exp(gamma)/Pi^2 (A335004).

cp's OEIS Frontend

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

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