A079938 -id:A079938 - cp's OEIS frontend

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

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

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

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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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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