A079939 -id:A079939 - cp's OEIS frontend

A079941 Greedy frac multiples of log(2): a(1)=1, Sum_{n>0} frac(a(n)*log(2)) = 1.

1, 3, 6, 13, 26, 39, 277, 642, 2291, 4582, 6231, 16402, 26573, 36744, 63317, 73488, 110232, 414355, 828710, 1206321, 2412642, 4410929, 5617250, 12026466, 31668469, 51310472, 70952475, 141904950, 394046381

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 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(4) = 13 since frac(1x) + frac(3x) + frac(6x) + frac(13x) < 1, while frac(1x) + frac(3x) + frac(6x) + frac(k*x) > 1 for all k>6 and k<13.

Cf. A079943 (denominators of convergents to ln2), A079934, A079939, A079940.

More terms from Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 29 2003

a(20)-a(29) from Sean A. Irvine, Aug 31 2025

A079937 Greedy frac multiples of Pi^2/6: a(1)=1, Sum_{n>=1} frac(a(n)*x) = 1 at x = Pi^2/6.

Original entry on oeis.org

1, 2, 14, 45, 107, 138, 276, 414, 1135, 2270, 6672, 12209, 18881, 180865, 361730, 542595, 723460, 2031679, 7945851, 15891702, 21805874, 29751725, 43611748, 65417622, 87223496, 362754007, 384559881, 406365755

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 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(4) = 45 since frac(1*x) + frac(2*x) + frac(14*x) + frac(45*x) < 1, while frac(1*x) + frac(2*x) + frac(14*x) + frac(k*x) > 1 for all k > 14 and k < 45.

Cf. A080017 (denominators of convergents to Pi^2/6), A079934, A079938, A079939.

a(15)-a(28) from Sean A. Irvine, Aug 30 2025

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

Original entry on oeis.org

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

A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 1.

Original entry on oeis.org

1, 2, 3, 8, 99, 33102, 66317, 265381, 1360120, 25510582, 78256779, 156513558, 209259755, 340262731, 1963319607, 6701487259, 8664806866, 13402974518, 20104461777, 26805949036, 33507436295, 40208923554, 46910410813

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 21 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(4) = 8 since frac(1x*) + frac(2*x) + frac(3*x) + frac(8*x) < 1, while frac(1*x) + frac(2*x) + frac(3*x) + frac(k*x) > 1 for all k > 3 and k < 8.

Cf. A002486 (denominators of convergents to Pi), A079934, A079937, A079939.

Digits := 100: a := []: s := 0: x := 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;

first(n)=my(v=vector(n),s=1.,p=Pi-3,k); for(m=1,oo, my(t=frac(p*m)); if(tCharles R Greathouse IV, Jul 25 2024

a(9) from Mark Hudson, Jan 30 2003

a(10)-a(23) from Charles R Greathouse IV, Jul 26 2024

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

nonn

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, etc.

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 *)

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

nonn

			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. Searching in the OEIS for "greedy frac" gives related sequences.

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, 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

Offset: 1

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

For definition of how the "Greedy Frac" sequence is defined, see other sequences in index.

			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. 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;

cp's OEIS Frontend

A079941 Greedy frac multiples of log(2): a(1)=1, Sum_{n>0} frac(a(n)*log(2)) = 1.

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A079937 Greedy frac multiples of Pi^2/6: a(1)=1, Sum_{n>=1} frac(a(n)*x) = 1 at x = Pi^2/6.

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

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A079938 Greedy frac multiples of Pi: a(1)=1, Sum_{n>=1} frac(a(n)*Pi) = 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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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, where "frac(y)" denotes the fractional part of y.

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