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.

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A349474 a(n) is the length of the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 1, 3, 3, 2, 3, 3, 4, 3, 2, 2, 7, 3, 4, 3, 3, 5, 3, 3, 2, 6, 3, 4, 1, 3, 2, 3, 2, 3, 4, 3, 8, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 5, 6, 4, 3, 4, 3, 4, 2, 5, 5, 7, 3, 3, 3, 5, 7, 7, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 7, 4, 4, 4, 3, 4, 6, 6, 3, 2, 4, 6, 3
Offset: 1

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Author

Amiram Eldar, Nov 19 2021

Keywords

Comments

a(n) = 1 if and only if n is a harmonic number (A001599).
a(n) <= 2 if and only if n is in A348865.

Examples

			a(1) = 1 since the harmonic mean of the divisors of 1 is 1 and its continued fraction has 1 element, {1}.
a(2) = 2 since the harmonic mean of the divisors of 2 is 4/3 = 1 + 1/3 and its continued fraction has 2 elements, {1, 3}.
a(4) = 4 since the harmonic mean of the divisors of 4 is 12/7 = 1 + 1/(1 + 1/(2 + 1/2)) and its continued fraction has 4 elements, {1, 1, 2, 2}.
		

Crossrefs

Row length of A349473.

Programs

  • Mathematica
    a[n_] := Length @ ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Array[a, 100]

A348866 Composite numbers k such that A099378(k) | (A099377(k) + 1).

Original entry on oeis.org

6, 15, 20, 28, 33, 35, 42, 51, 66, 69, 70, 84, 87, 114, 117, 123, 135, 138, 140, 141, 153, 159, 177, 186, 204, 207, 210, 213, 249, 258, 267, 270, 273, 276, 282, 285, 297, 303, 308, 321, 339, 348, 354, 357, 372, 393, 399, 402, 411, 420, 426, 432, 435, 447, 464
Offset: 1

Views

Author

Amiram Eldar, Nov 02 2021

Keywords

Comments

A disjoint union of the harmonic numbers (A001599) and the composite numbers whose harmonic mean of divisors is of the form m - 1/k, where m and k are integers.
If p is an odd prime, then the harmonic mean of its divisors is p*tau(p)/sigma(p) = p*A000005(p)/A000203(p) = p/((p+1)/2), so A099378(p) | (A099377(p) + 1). Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, it includes all the semiprimes of the form 3*p, where p == 2 (mod 3).

Examples

			15 is a term since it is composite, the harmonic mean of divisors of 15 is 5/2 and 2 | (5+1).
		

Crossrefs

Programs

  • Mathematica
    h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := Divisible[Numerator[(h1 = h[n])] + 1, Denominator[h1]]; Select[Range[1000], CompositeQ[#] && q[#] &]
Showing 1-2 of 2 results.