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.

A284648 Numerator of sum of reciprocals of all divisors of all positive integers <= n.

Original entry on oeis.org

1, 5, 23, 67, 407, 527, 4169, 9913, 33379, 7583, 89461, 102397, 1408777, 1532329, 8238221, 17872837, 316811189, 343357709, 6768841271, 7257705647, 7612437167, 7993370447, 189434541721, 202820113921, 1047296788661, 1090542483461, 3390610314383, 3551237180783, 105395281238707
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 31 2017

Keywords

Comments

The value of (1/n)*Sum_{k=1..n} sigma(k)/k approaches Pi^2/6.

Examples

			1, 5/2, 23/6, 67/12, 407/60, 527/60, 4169/420, 9913/840, 33379/2520, 7583/504, 89461/5544, 102397/5544, 1408777/72072, 1532329/72072, 8238221/360360, ...

References

  • József Sándor, Dragoslav S. Mitrinović, and Borislav Crstici, Handbook of Number Theory I, Springer, 2006, Section III.5, p. 82.
  • Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 99.

Links

Crossrefs

Cf. A000203, A017665, A017666, A108775, A284650 (denominators).

Programs

  • Maple
    with(numtheory): seq(numer(add(sigma(k)/k, k=1..n)), n=1..40); # Ridouane Oudra, Jan 21 2024
  • Mathematica
    Table[Numerator[Sum[DivisorSigma[-1, k], {k, 1, n}]], {n, 1, 29}]
Table[Numerator[Sum[DivisorSigma[1, k]/k, {k, 1, n}]], {n, 1, 29}]
nmax = 29; Rest[Numerator[CoefficientList[Series[1/(1 - x) Sum[Log[1/(1 - x^k)], {k, 1, nmax}], {x, 0, nmax}], x]]]
  • PARI
    for(n=1, 29, print1(numerator(sum(k=1, n, sigma(k)/k)),", ")) \\ Indranil Ghosh, Mar 31 2017
  • Python
    from sympy import divisor_sigma, Integer
print([sum(divisor_sigma(k)/Integer(k) for k in range(1, n + 1)).numerator for n in range(1, 30)]) # Indranil Ghosh, Mar 31 2017

Formula

G.f.: (1/(1 - x))*Sum_{k>=1} log(1/(1 - x^k)) (for a(n)/A284650(n), see example).
a(n) = numerator of Sum_{k=1..n} Sum_{d|k} 1/d.
a(n) = numerator of Sum_{k=1..n} sigma(k)/k.
a(n) = numerator of Sum_{k=1..n} floor(n/k)/k. - Ridouane Oudra, Jan 21 2024

A330899 Numbers m such that (1/m) * Sum_{k=1..m} sigma(k)/k sets a record value, where sigma(k) is the sum of divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 36, 42, 48, 56, 60, 72, 84, 90, 96, 100, 108, 112, 120, 144, 156, 168, 180, 192, 210, 240, 276, 280, 288, 300, 312, 324, 330, 336, 360, 396, 408, 420, 480, 528, 540, 576, 600, 630, 660, 672, 720, 756, 792
Offset: 1

Views

Author

Amiram Eldar, May 01 2020

Keywords

Comments

Numbers m such that the mean of the abundancy index sigma(k)/k in the range 1..m is closer to the asymptotic mean Pi^2/6 than the mean in any smaller range.
Since (1/m) * Sum_{k=1..m} sigma(k)/k < Pi^2/6 for all m, and the limit is Pi^2/6 as m -> infinity, this sequence is infinite.

Examples

			The mean abundancy in the range 1..m for m = 1, 2, ..., 6 is 1, 1.25, 1.277..., 1.395..., 1.356..., 1.463..., so the record values occur at 1, 2, 3, 4 and 6.

Links

Crossrefs

Cf. A000203, A013661, A284648, A284650.

Programs

  • Mathematica
    seq = {}; s = 0; rm = 0; Do[s += DivisorSigma[1, n]/n; r = s/n; If[r > rm, rm = r; AppendTo[seq, n]], {n, 1, 1000}]; seq

A380315 Denominator of sum of reciprocals of all prime divisors of all positive integers <= n.

Original entry on oeis.org

1, 2, 6, 3, 15, 30, 210, 105, 35, 70, 770, 1155, 15015, 30030, 30030, 15015, 255255, 170170, 3233230, 1616615, 4849845, 9699690, 223092870, 111546435, 22309287, 44618574, 14872858, 7436429, 215656441, 6469693230, 200560490130, 100280245065, 100280245065
Offset: 1

Views

Author

Ilya Gutkovskiy, Jan 20 2025

Keywords

Comments

Prime divisors counted without multiplicity.
Differs from A379370 first at n=15.

Examples

			0, 1/2, 5/6, 4/3, 23/15, 71/30, 527/210, 316/105, 117/35, 283/70, 3183/770, 5737/1155, 75736/15015, ...

Crossrefs

Cf. A000720, A007947, A013939, A024924, A028235, A284650, A379370, A379368, A380314 (numerators).

Programs

  • Mathematica
    Table[DivisorSum[n, 1/# &, PrimeQ[#] &], {n, 1, 33}] // Accumulate // Denominator
Table[Sum[Floor[n/Prime[k]]/Prime[k], {k, 1, n}], {n, 1, 33}] // Denominator
nmax = 33; CoefficientList[Series[1/(1 - x) Sum[x^Prime[k]/(Prime[k] (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
  • PARI
    a(n) = my(vp=primes(primepi(n))); denominator(sum(k=1, #vp, (n\vp[k])/vp[k])); \\ Michel Marcus, Jan 26 2025

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^prime(k) / (prime(k)*(1 - x^prime(k))).
a(n) is the denominator of Sum_{k=1..pi(n)} floor(n/prime(k)) / prime(k).

A384818 Denominator of the sum of the reciprocals of all square divisors of all positive integers <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 4, 4, 2, 18, 18, 18, 36, 36, 36, 36, 144, 144, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 1200, 1200, 1200, 1200, 1200, 600, 600, 600, 600, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 1800, 600, 600, 600, 1200, 58800, 58800, 58800, 58800, 58800
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 10 2025

Keywords

Examples

			1, 2, 3, 17/4, 21/4, 25/4, 29/4, 17/2, 173/18, 191/18, 209/18, 463/36, ...

Crossrefs

Cf. A007407, A017668, A284650, A309125, A373440, A384817 (numerators).

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/(1 - x) Sum[x^(k^2)/(k^2 (1 - x^(k^2))), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
Table[Sum[Floor[n/k^2]/k^2, {k, 1, Floor[Sqrt[n]]}], {n, 1, 53}] // Denominator
  • PARI
    a(n) = denominator(sum(k=1, n, sumdiv(k, d, if (issquare(d), 1/d)))); \\ Michel Marcus, Jun 10 2025

Formula

G.f. for fractions: (1/(1 - x)) * Sum_{k>=1} x^(k^2) / (k^2*(1 - x^(k^2))).
a(n) is the denominator of Sum_{k=1..floor(sqrt(n))} floor(n/k^2) / k^2.
A384817(n) / a(n) ~ Pi^4 * n / 90.
Showing 1-4 of 4 results.

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