A367987 -id:A367987 - cp's OEIS frontend

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

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

Offset: 1

Amiram Eldar, Jun 21 2020

The asymptotic mean of A367987. - Amiram Eldar, Dec 23 2023

			1.450032145362120831608395887189223422325062117447167...

Cf. A000082, A001615, A013661, A065465, A065484, A367987.

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{-1, 1, 2, 0, -1}, {0, 2, -3, 6, -5}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

prodeulerrat(1 + p/((p-1)*(p+1)^2)) \\ Amiram Eldar, Mar 18 2021

Equals Sum_{k>=1} 1/A000082(k) = Sum_{k>=1} 1/(k * A001615(k)).

Equals A013661 * A065465. - Amiram Eldar, Dec 23 2023

More digits from Vaclav Kotesovec, Sep 19 2020

A367988 The sum of the divisors of the square root of the largest unitary divisor of n that is a square.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 1, 3, 1, 1, 1, 7, 1, 4, 1, 3, 1, 1, 1, 1, 6, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 7, 8, 6, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 4, 15, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 6, 3, 1, 1, 1, 7, 13, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000203, A071974, A268335, A351568, A367987.

Cf. A065463.

f[p_, e_] := If[EvenQ[e], (p^(e/2 + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = (p^(e/2+1)-1)/(p-1) if e is even and 1 otherwise.
a(n) = A000203(A071974(n)).
a(n) >= 1, with equality if and only if n is an exponentially odd number (A268335).
Dirichlet g.f.: zeta(2*s) * zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1)).
From Vaclav Kotesovec, Apr 20 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/((p^s + 1)*p^(2*s - 1))).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * f(s).
Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 3*gamma - 1 + f'(1)/f(1)) / 2, where
f(1) = A065463 = Product_{p prime} (1 - 1/(p*(1+p))) = 0.704442200999165592736603350326637210188586431417098049414226842591097056682...
f'(1) = f(1) * Sum_{p prime} (3*p+2)*log(p)/((p+1)*(p^2+p-1)) = f(1) * 1.167129912223800181472507785468113632129480568043855995406075158923507536957...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

A335762 Decimal expansion of Product_{p prime} (1 + p/((p-1)*(p+1)^2)).

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