cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 19 2024

Keywords

Comments

First differs from A318491 at n = 27.
The abundancy index of a number k is sigma(k)/k = A017665(k)/A017666(k).

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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]
  • PARI
    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)));}

Formula

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).

A318491 a(n) is the numerator of Sum_{d|n} Sum_{j|d} 1/j.

Original entry on oeis.org

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, 142, 255, 59, 77, 63, 321, 161, 175, 33, 289, 75, 195, 63, 539, 83, 25, 87, 391, 374, 235, 95, 301, 162, 43, 245, 459, 107, 355, 23, 105, 91, 295, 119, 1309, 123, 315, 170, 769, 297
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 27 2018

Keywords

Examples

			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, ...

Links

Crossrefs

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A318492 (denominators).
Cf. A001620, A013661, A306016.

Programs

  • Maple
    f:= proc(n) local d;
   numer(add(numtheory:-sigma(d)/d, d = numtheory:-divisors(n))) end proc:
map(f, [$1..65]); # Robert Israel, Jan 13 2025
  • Mathematica
    Numerator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 65}]]
Numerator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 65}]]
nmax = 65; Rest[Numerator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 65; Rest[Numerator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]
  • PARI
    a(n) = numerator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Formula

Numerators 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).
Numerators 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) = numerator of Sum_{d|n} sigma(d)/d.
a(n) = numerator of (1/n)*Sum_{d|n} d*tau(d).
If p is a prime, a(p) = 2*p + 1.
Sum_{k=1..n} a(k)/A318492(k) ~ zeta(2) * n * (log(n) + 2*gamma - 1 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 25 2024

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

Views

Author

Jaroslav Krizek, Dec 23 2018

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Magma
    [Numerator(&*[&+[c: c in Divisors(d)] / d: d in Divisors(n)]): n in [1..100]];
  • Mathematica
    Array[Numerator@ Product[DivisorSigma[1, d]/d, {d, Divisors@ #}] &, 51] (* Michael De Vlieger, Jan 19 2019 *)
  • PARI
    a(n) = my(d=divisors(n)); numerator(prod(k=1, #d, sigma(d[k])/d[k])); \\ Michel Marcus, Dec 23 2018, May 11 2020

Formula

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

Views

Author

Jaroslav Krizek, Jan 21 2019

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Magma
    [Denominator(&*[&+[c: c in Divisors(d)] / d: d in Divisors(n)]): n in [1..100]];
  • Mathematica
    Array[Denominator@ Product[DivisorSigma[1, d]/d, {d, Divisors@ #}] &, 51]
  • PARI
    a(n) = my(p=1); fordiv(n, d, p *= sigma(d)/d); denominator(p); \\ Michel Marcus, Jan 21 2019

Formula

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)).
Showing 1-4 of 4 results.

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