A318491 -id:A318491 - cp's OEIS frontend

A374777 Numerator of the mean abundancy index of the divisors of n.

1, 5, 7, 17, 11, 35, 15, 49, 34, 11, 23, 119, 27, 75, 77, 129, 35, 85, 39, 187, 5, 115, 47, 343, 86, 135, 71, 85, 59, 77, 63, 107, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 187, 235, 95, 301, 54, 43, 245, 153, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315

Offset: 1

Amiram Eldar, Jul 19 2024

First differs from A318491 at n = 27.

The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k).

			For n = 2, n has 2 divisors, 1 and 2. Their abundancy indices are sigma(1)/1 = 1 and sigma(2)/2 = 3/2, and their mean abundancy index is (1 + 3/2)/2 = 5/4. Therefore a(2) = numerator(5/4) = 5.

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

Cf. A000005, A000203, A013661, A017665, A017666, A038040, A060640, A245214, A318491, A318492, A374778 (denominators), A374779, A374780, A374781.

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

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

Let f(n) = a(n)/A374778(n). Then:
f(n) = (Sum_{d|n} sigma(d)/d) / tau(n), where sigma(n) is the sum of divisors of n (A000203), and tau(n) is their number (A000005).
f(n) is multiplicative with f(p^e) = ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2).
f(n) = A318491(n)/(A318492(n)*A000005(n)).
f(n) = (Sum_{d|n} d*tau(d)) / (n*tau(n)) = A060640(n)/A038040(n).
Dirichlet g.f. of f(n): zeta(s) * Product_{p prime} ((p/(p-1)^2) * ((p^s-1)*log((1-1/p^s)/(1-1/p^(s+1))) + p-1)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = Product_{p prime} ((p/(p-1)) * (1 - log(1 + 1/p))) = 1.3334768464... . For comparison, the asymptotic mean of the abundancy index over all the positive integers is zeta(2) = 1.644934... (A013661).
Lim sup_{n->oo} f(n) = oo (i.e., f(n) is unbounded).

A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 18, 37, 38, 13, 40, 41, 2, 43, 44, 45, 46, 47, 16, 49, 5, 51, 52, 53, 27, 5, 8, 19, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 4, 69, 14, 71, 36, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Aug 27 2018

			1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A068986 (positions of 1's), A318491 (numerators).

Denominator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 75}]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

a(n) = denominator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Denominators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).
Denominators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).
a(n) = denominator of Sum_{d|n} sigma(d)/d.
a(n) = denominator of (1/n)*Sum_{d|n} d*tau(d).

A322673 a(n) = numerator of Product_{d|n} (sigma(d)/d) where sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 3, 4, 21, 6, 4, 8, 315, 52, 81, 12, 49, 14, 144, 64, 9765, 18, 338, 20, 11907, 1024, 324, 24, 1225, 186, 441, 2080, 72, 30, 20736, 32, 615195, 256, 729, 2304, 753571, 38, 900, 3136, 321489, 42, 65536, 44, 11907, 21632, 1296, 48, 1177225, 456, 233523, 576

Offset: 1

Jaroslav Krizek, Dec 23 2018

			For n=4; Product_{d|4} (sigma(d)/d) = (sigma(1)/1)*(sigma(2)/2)*(sigma(4)/4) = (1/1)*(3/2)*(7/4) = 21/8; a(4) = 21.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A318491, A318492, A299788 (denominator).

[Numerator(&*[&+[c: c in Divisors(d)] / d: d in Divisors(n)]): n in [1..100]];

Array[Numerator@ Product[DivisorSigma[1, d]/d, {d, Divisors@ #}] &, 51] (* Michael De Vlieger, Jan 19 2019 *)

a(n) = my(d=divisors(n)); numerator(prod(k=1, #d, sigma(d[k])/d[k])); \\ Michel Marcus, Dec 23 2018, May 11 2020

a(n) = n + 1 for n = primes (A000040).

A299788 a(n) = denominator of Product_{d|n} (sigma(d)/d) where sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 2, 3, 8, 5, 1, 7, 64, 27, 25, 11, 3, 13, 49, 25, 1024, 17, 27, 19, 1000, 441, 121, 23, 16, 125, 169, 729, 7, 29, 625, 31, 32768, 121, 289, 1225, 5832, 37, 361, 1521, 6400, 41, 2401, 43, 1331, 3375, 529, 47, 3072, 343, 31250, 289, 17576, 53, 6561, 3025, 49

Offset: 1

Jaroslav Krizek, Jan 21 2019

			For n=4; Product_{d|4} (sigma(d)/d) = (sigma(1)/1)*(sigma(2)/2)*(sigma(4)/4) = (1/1)*(3/2)*(7/4) = 21/8; a(4) = 8.

Cf. A000203, A318491, A318492, A322673 (numerator).

[Denominator(&*[&+[c: c in Divisors(d)] / d: d in Divisors(n)]): n in [1..100]];

Array[Denominator@ Product[DivisorSigma[1, d]/d, {d, Divisors@ #}] &, 51]

a(n) = my(p=1); fordiv(n, d, p *= sigma(d)/d); denominator(p); \\ Michel Marcus, Jan 21 2019

a(n) = 1 for n = 1, 6, ... (no other n <= 5*10^6).
a(n) = n for n = primes (A000040).
a(n) = n * sqrt(n) for n in A280076 (union of 1 and squares of primes (A001248)).

cp's OEIS Frontend

A374777 Numerator of the mean abundancy index of the divisors of n.

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A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

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A322673 a(n) = numerator of Product_{d|n} (sigma(d)/d) where sigma(k) = the sum of the divisors of k (A000203).

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A299788 a(n) = denominator of Product_{d|n} (sigma(d)/d) where sigma(k) = the sum of the divisors of k (A000203).

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