A322887 -id:A322887 - cp's OEIS frontend

A061742 a(n) is the square of the product of first n primes.

1, 4, 36, 900, 44100, 5336100, 901800900, 260620460100, 94083986096100, 49770428644836900, 41856930490307832900, 40224510201185827416900, 55067354465423397733736100, 92568222856376731590410384100, 171158644061440576710668800200900

Offset: 0

Jason Earls, Jun 21 2001

nonn

Squares of primorials (first definition, A002110).
Exponential superabundant numbers: numbers k with a record value of the exponential abundancy index, A051377(k)/k > A051377(m)/m for all m < k. - Amiram Eldar, Apr 13 2019
Numbers k with a record value of A056170(k), or least number k with A056170(k) = n. - Amiram Eldar, Apr 15 2019
Empirically, these are possibly the denominators for 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2. The numerators are listed in A136370. - Petros Hadjicostas, May 14 2020
a(n) = least k such that rad(k/rad(k)) = A002110(n). - David James Sycamore, Jun 10 2024

			a(4) = 2^2 * 3^2 * 5^2 * 7^2 = 44100.

Harry J. Smith, Table of n, a(n) for n = 0..100

Equals A002110^2.
Cf. A001248, A051377, A056170, A129575, A136370, A322887.
Cf. A007947.

[n eq 0 select 1 else (&*[NthPrime(j)^2: j in [1..n]]): n in [0..20]]; // G. C. Greubel, Apr 19 2019

a:= proc(n) option remember; `if`(n=0, 1, ithprime(n)^2*a(n-1)) end:
seq(a(n), n=0..15);  # Alois P. Heinz, May 14 2020

a[n_]:=Product[Prime[i]^2, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

for(n=0,20,print1(prod(k=1,n, prime(k)^2), ", "))

{ n=-1; m=1; forprime (p=2, prime(101), write("b061742.txt", n++, " ", m^2); m*=p ) } \\ Harry J. Smith, Jul 27 2009

[product(nth_prime(j)^2 for j in (1..n)) for n in (0..20)] # G. C. Greubel, Apr 19 2019

a(n) = Product_{j=1..n} A001248(j). - Alois P. Heinz, May 14 2020

a(n) = A228593(n) * A000040(n), for n>0. - Marco Zárate, Jun 11 2024

A306633 Decimal expansion of zeta(2)/zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 02 2019

Equals the asymptotic mean of the unitary abundancy index, lim_{n->oo} (1/n) * Sum{k=1..n} usigma(k)/k, where usigma(k) is the sum of the unitary divisors of k (A034448).
From Amiram Eldar, May 12 2023: (Start)
Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
In general, the asymptotic mean of the abundancy index of the k-free numbers (numbers that are not divisible by a k-th power other than 1) is zeta(2)/zeta(k+1) (Jakimczuk and Lalín, 2022). (End)

			1.3684327776202058757367658539847919411308139146524...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.

RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]

zeta(2)/zeta(3) \\ Michel Marcus, Mar 04 2019

Equals A013661/A002117 = 1/A253905.
Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - Amiram Eldar, Jun 23 2020
Equals Product_{p prime} (1 + 1/(p*(p+1))). - Amiram Eldar, Aug 10 2020
Equals Sum_{k>=1} mu(k)^2/(k*psi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Dedekind psi function values, A001615). - Amiram Eldar, Aug 18 2020

A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Sep 27 2019

			1.252707785375446126053750751934283060439237967108915...

László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294.

Cf. A000010, A072911, A322887, A327837, A361013.

$MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e * (EulerPhi[e] - EulerPhi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

Equals lim_{m->oo} (1/m) Sum_{k=1..m} A072911(k).

Equals Product_{p prime} (1 + Sum_{e >= 3} (phi(e) - phi(e-1))/p^e), where phi is the Euler totient function (A000010).

A332889 a(n) = number of strict partition numbers >1 that are proper divisors of the n-th strict partition number.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 2, 2, 4, 2, 3, 1, 1, 3, 1, 1, 5, 4, 3, 0, 3, 1, 1, 3, 8, 3, 5, 3, 4, 6, 5, 6, 3, 2, 7, 10, 1, 1, 9, 2, 4, 3, 7, 11, 3, 6, 9, 1, 0, 1, 9, 3, 3, 2, 1, 6, 11, 8, 2, 1, 7, 2, 6, 2, 4, 12, 3, 0, 4, 8, 4, 4, 1, 7, 0, 1, 9, 7, 5, 5, 1, 1, 6, 5, 4

Offset: 3

Clark Kimberling, Mar 11 2020

nonn

Let p(n) = number of strict partitions of n. Then p(11) = 12, which is divisible by these 6 strict partition numbers: p(2) = 1, p(3) = 2, p(5) = 3, p(6) = 4, p(8) = 6, and p(11) = 12; discounting 1 and 12 leaves a(11) = 4 divisors.

Cf. A000009 (strict partition numbers), A322887, A332888.

p[n_] := PartitionsQ[n]; t[n_] := Table[p[k], {k, 0, n}]
-2+Table[Length[Intersection[t[n], Divisors[p[n]]]], {n, 3, 130}]

a(n) = A332888(n) - 2 for n >= 3.

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A061742 a(n) is the square of the product of first n primes.

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A306633 Decimal expansion of zeta(2)/zeta(3).

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A327838 Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

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A332889 a(n) = number of strict partition numbers >1 that are proper divisors of the n-th strict partition number.

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