A106394 -id:A106394 - cp's OEIS frontend

A105401 a(n) is the final term of the n-th row of A106394.

1, 2, 3, 12, 30, 180, 395780, 56, 5339880, 18448242120, 1192281609186360, 180180, 393869810468558572779517216129800, 2914872400440, 2914872400440, 180026486640, 4193048661254917704240, 2640766128, 11832093494672880

Offset: 1

Leroy Quet, May 01 2005

It's interesting that some terms are so much smaller than their neighbors, like row 8 (1 1 2 5 56) and row 26 (1 1 1 2 3 48 3952 76271067 17451826907684400). - Joshua Zucker, May 11 2006

Amiram Eldar, Table of n, a(n) for n = 1..31

Cf. A001008, A002805, A106394, A112330.

egyptFraction[f_] := Ceiling[1/Most[NestWhileList[# - 1/Ceiling[1/#] &, f, # != 0 &]]]; a[n_] := egyptFraction[HarmonicNumber[n]][[-1]]; Array[a, 20] (* Amiram Eldar, Apr 09 2022 *)

More terms from Joshua Zucker, May 11 2006

A112330 a(n) is the number of terms in the n-th row of A106394.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 5, 7, 8, 7, 6, 9, 7, 8, 8, 9, 8, 8, 9, 8, 11, 11, 11, 11, 9, 14, 10, 15, 12, 12, 15, 11, 14, 10, 14, 12, 13, 12, 16, 15, 13, 12, 12, 17, 13, 16, 16, 15, 16, 19, 14, 17, 16, 16, 21, 15, 18, 16, 18, 18, 18, 19, 21, 20, 19, 20, 22, 17, 20, 27, 19, 25, 20, 18, 23, 24

Offset: 1

Leroy Quet, Sep 04 2005

nonn

			H(4) = 1 + 1/2 + 1/3 + 1/4 = 25/12 has the Egyptian fraction expansion, by the greedy algorithm, of 1 + 1 + 1/12. Since there are 3 terms in this expansion, a(4) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..142

Cf. A001008, A002805, A105401, A106394, A106395.

egyptFraction[f_] := Ceiling[1/Most[NestWhileList[# - 1/Ceiling[1/#] &, f, # != 0 &]]]; a[n_] := Length[egyptFraction[HarmonicNumber[n]]]; Array[a, 100] (* Amiram Eldar, Apr 09 2022 *)

More terms from David Wasserman, Apr 16 2009

A106395 a(n) = sum of terms of n-th row of A106394.

Original entry on oeis.org

1, 3, 6, 14, 36, 194, 396309, 65, 5340390, 18448355363, 1192281639870330, 180505, 393869810468558586067193126963349, 2914872842753, 2914872842768, 180026806679, 4193048661303182427879, 2640786324, 11832093582726989

Offset: 1

Leroy Quet, May 01 2005

nonn

			By the greedy algorithm, sum{k=1 to 4} 1/k = 1 + 1 + 1/12.
So a(4) = 1+1+12 = 14.

Cf. A106394.

More terms from Jud McCranie, May 03 2005

More terms from Joshua Zucker, May 11 2006

A364200 Minimal number of terms of mixed-sign Egyptian fraction f such that H(n) + f is an integer, where H(n) is the n-th harmonic number.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 3, 2, 2, 3, 4, 3, 4, 4, 4, 4, 4, 5, 5, 4, 5, 5, 5, 4, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6

Offset: 1

Denis Ivanov, Jul 13 2023

For H(n) - floor(H(n)) and ceiling(H(n)) - H(n), the shortest mixed-sign Egyptian fractions are calculated, and the smaller length of fractions is selected.
Similar to A106394 and A224820. But those sequences use the greedy algorithm, which does not guarantee the shortest length of expansion.
For 1 < n < 41, a(n) < A363937(n) only for n = 10 and n = 22.

			For n=10: H(10) = 7381/2520 = 2.928...; H(10) - floor(H(10)) = 7381/2520 - 2 = 2341/2520 = 1/2 + 1/7 + 1/8 + 1/9 + 1/20, which cannot be expressed as the sum of fewer than 5 reciprocals, and ceiling(H(10)) - H(10) = 3 - 7381/2520 = 179/2520 = 1/30 + 1/42 + 1/72, which cannot be expressed as the sum of fewer than 3 reciprocals, so A363937(10) = 3.
But 179/2520 = 1/14 - 1/2520 (a "mixed-sign Egyptian fraction"), so a(10) = 2.

Ron Knott, Egyptian Fractions

Cf. A363937.

check[f_, k_] := (If[Numerator@f == 1, Return@True];
   If[k == 1, Return@False];
   Catch[Do[If[check[f - 1/i, k - 1], Throw@True],
     {i, Range[Ceiling[1/f], Floor[k/f]]}];
    Throw@False]);
checkMixed[f_, k_, m_] := If[m == 1,
   Catch[Do[If[check[1/i - f, k], Throw@True],
     {i, Range[2, Floor[1/f]]}];
    Throw@False],
   checkMixed[f, k, m - 1]];
a[n_] := (h = HarmonicNumber[n];
  d = Min[h - Floor@h, Ceiling@h - h];
  j = 1;
  While[Not@check[d, j], j++];
  res = j;
  Do[
   If[checkMixed[d, i - m, m], res = i],
   {i, 2, j - 1}, {m, 1, i - 1}];
  res);

a(n) <= A363937(n).

cp's OEIS Frontend

A105401 a(n) is the final term of the n-th row of A106394.

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A112330 a(n) is the number of terms in the n-th row of A106394.

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A106395 a(n) = sum of terms of n-th row of A106394.

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A364200 Minimal number of terms of mixed-sign Egyptian fraction f such that H(n) + f is an integer, where H(n) is the n-th harmonic number.

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