A079941 -id:A079941 - cp's OEIS frontend

A079940 Greedy fractional multiples of 1/e: a(1)=1, Sum_{n>0} frac(a(n)/e) = 1.

1, 3, 4, 11, 87, 193, 386, 579, 1457, 23225, 49171, 98342, 147513, 196684, 566827, 13580623, 28245729, 56491458, 84737187, 112982916, 438351041, 466596770, 494842499

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 2003

The n-th greedy fractional multiple of x is the smallest integer m that does not cause Sum_{k=1..n} frac(m*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of 1/e.

After a(20), there is only 109305220 - 297122396/e = ~1.06317354345346734...*10^-8 to work with.

			a(4) = 11 since frac(1x) + frac(3x) + frac(4x) + frac(11x) < 1, while frac(1x) + frac(3x) + frac(4x) + frac(k*x) > 1 for all k>4 and k<11.

K. Girstmair, On the Asymptotic Behavior of Dedekind Sums, J. Int. Seq. 17 (2014) # 14.7.7, example 2.

Cf. A007676 (numerators of convergents to e), A079934, A079939, A079941.

Digits := 100: a := []: s := 0: x := 1.0/exp(1.0): for n from 1 to 1000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, ps = Plus @@ Table[ FractionalPart[ a[i]*E^-1], {i, n - 1}]}, While[ ps + FractionalPart[k*E^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 20}] (* Robert G. Wilson v, Nov 03 2004 *)

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003
a(16)-a(20) from Robert G. Wilson v, Nov 03 2004
a(21)-a(23) from Sean A. Irvine, Aug 30 2025

A080142 Greedy frac multiples of 1/Pi: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=1/Pi, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 2, 22, 44, 66, 88, 110, 355, 710, 1065, 1420, 1775, 2130, 2485, 2840, 3195, 3550, 3905, 4260, 4615, 4970, 5325, 5680, 6035, 6390, 6745, 7100, 7455, 7810, 8165, 104348, 104703, 105058, 105413, 105768, 208696, 209051, 312689, 313044, 417037

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 30 2003

The n-th greedy frac multiple of x is the smallest integer that does not cause Sum_{k=1..n} frac(a(k)*x) to exceed unity; an infinite number of terms appear as the denominators of the convergents to the continued fraction of x.

			a(3) = 22 since frac(1x) + frac(2x) + frac(22x) < 1, while frac(1x) + frac(2x) + frac(k*x) > 1 for all k>2 and k<22.

Cf. A079938, A079939, A079940, A079941.

Digits := 1000: a := []: s := 0: x := evalf(1.0/Pi): for n from 1 to 10000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

a[1] = 1; a[n_] := a[n] = Block[{k = a[n - 1] + 1, fps = Plus @@ Table[FractionalPart[a[i]*Pi^-1], {i, n - 1}]}, While[fps + FractionalPart[k*Pi^-1] > 1, k++ ]; a[n] = k]; Do[ Print[ a[n]], {n, 40}] (* Robert G. Wilson v, Nov 03 2004 *)

A079943 Denominators of the convergents of the continued fraction for log(2).

Original entry on oeis.org

1, 3, 10, 13, 88, 277, 365, 642, 1649, 2291, 3940, 6231, 10171, 36744, 377611, 414355, 791966, 1206321, 3204608, 4410929, 7615537, 12026466, 19642003, 70952475, 161546953, 555593334, 717140287, 9878417065, 69866059742, 289342656033

Offset: 1

Paul D. Hanna, Jan 19 2003

			9/13 = 0.6923076923076923...
61/88 = 0.69318181818181818...
192/277 = 0.6931407942238267148...
...
log(2) = 0.6931471805599453...

Cf. A002162 (decimal expansion of log(2)), A079941, A079942 (numerators).

A079942 Numerators of the convergents of the continued fraction for log(2).

Original entry on oeis.org

0, 1, 2, 7, 9, 61, 192, 253, 445, 1143, 1588, 2731, 4319, 7050, 25469, 261740, 287209, 548949, 836158, 2221265, 3057423, 5278688, 8336111, 13614799, 49180508, 111975815, 385107953, 497083768, 6847196937, 48427462327, 200557046245, 248984508572, 449541554817

Offset: 1

Paul D. Hanna, Jan 19 2003

			9/13 = 0.6923076923076923...
61/88 = 0.69318181818181818...
192/277 = 0.6931407942238267148...
...
log(2) = 0.6931471805599453...

Cf. A002162 (decimal expansion of log(2)), A079941, A079943 (denominators).

Convergents[ContinuedFraction[Log[2],40]]//Numerator (* Harvey P. Dale, Jun 05 2021 *)

Corrected and extended by Harvey P. Dale, Jun 05 2021

A080157 Greedy frac multiples of gamma: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=gamma, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 2, 7, 9, 26, 52, 149, 272, 395, 790, 1185, 1580, 5653, 10911, 16169, 26685, 58628, 85313, 117256, 175884, 559595, 2179752, 5420066

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

			a(3) = 7 since frac(1x) + frac(2x) + frac(7x) < 1, while frac(1x) + frac(2x) + frac(k*x) > 1 for all k>2 and k<7.

Cf. A079938, A079939, A079940, A079941, A080142.

Digits := 1000: a := []: s := 0: x := evalf(gamma): for n from 1 to 10000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

A080158 Greedy frac multiples of Catalan's constant, G: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=G=A006752, where "frac(y)" denotes the fractional part of y.

Original entry on oeis.org

1, 11, 107, 10579, 21158, 53014, 106028, 625708, 721157, 1442314, 2163471, 2884628, 3605785, 4326942, 16682060, 66007083, 115332106, 164657129, 329314258, 493971387, 658628516

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 31 2003

For definition of how the "Greedy Frac" sequence is defined, see A079938.

			a(3) = 107 since frac(1x) + frac(11x) + frac(107x) < 1, while frac(1x) + frac(11x) + frac(k*x) > 1 for all k>11 and k<107.

Cf. A006752, A079938, A079939, A079940, A079941, A080142, A080157.

Digits := 1000: a := []: s := 0: x := evalf(Catalan): for n from 1 to 5000000 do: temp := evalf(s+frac(n*x)): if (temp<1.0) then a := [op(a),n]: print(n): s := s+evalf(frac(n*x)): fi: od: a;

a(15)-a(21) from Sean A. Irvine, Sep 05 2025

cp's OEIS Frontend

A079940 Greedy fractional multiples of 1/e: a(1)=1, Sum_{n>0} frac(a(n)/e) = 1.

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A080142 Greedy frac multiples of 1/Pi: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=1/Pi, where "frac(y)" denotes the fractional part of y.

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A079943 Denominators of the convergents of the continued fraction for log(2).

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A079942 Numerators of the convergents of the continued fraction for log(2).

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A080157 Greedy frac multiples of gamma: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=gamma, where "frac(y)" denotes the fractional part of y.

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A080158 Greedy frac multiples of Catalan's constant, G: a(1)=1, Sum_{n>0} frac(a(n)*x) = 1 at x=G=A006752, where "frac(y)" denotes the fractional part of y.

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