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.

Previous Showing 21-26 of 26 results.

A076797 Greedy powers of (Pi/5): Sum_{n>=1} (Pi/5)^a(n) = 1.

Original entry on oeis.org

1, 3, 5, 8, 15, 17, 20, 25, 28, 30, 32, 35, 43, 54, 58, 65, 67, 70, 73, 76, 82, 86, 89, 94, 97, 100, 107, 112, 119, 121, 124, 130, 133, 135, 137, 141, 143, 146, 153, 156, 163, 166, 169, 175, 177, 180, 185, 195, 199, 204, 210, 212, 217, 220, 226, 229, 234, 239
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity.

Examples

			Pi/5 + (Pi/5)^3 + (Pi/5)^5 < 1 and Pi/5 + (Pi/5)^3 + (Pi/5)^4 > 1; since the power 4 makes the sum > 1, 5 is the 3rd greedy power of (Pi/5), so a(3)=5.

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(Pi / 5.): pexp := p: a := []: for i from 1 to 800 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;

Formula

a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1} = log_x(x^frac(g_n) - x) (n > 0) at x = Pi/5 and frac(y) = y - floor(y).

Extensions

Corrected by T. D. Noe, Nov 02 2006

A076798 Greedy powers of (Pi/6): Sum_{n>=1} (Pi/6)^a(n) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 10, 11, 12, 14, 20, 21, 22, 25, 28, 33, 35, 37, 38, 39, 44, 45, 47, 49, 50, 52, 55, 56, 58, 59, 61, 63, 64, 71, 72, 78, 83, 84, 85, 88, 89, 93, 94, 96, 98, 100, 101, 104, 105, 106, 109, 114, 116, 117, 120, 121, 122, 125, 133, 134, 138, 140, 141, 142
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series sum_{k=1..n} x^a(k) to exceed unity.

Examples

			a(4)=5 since (Pi/6) +(Pi/6)^2 +(Pi/6)^3 +(Pi/6)^5 < 1 and (Pi/6) +(Pi/6)^2 +(Pi/6)^3 +(Pi/6)^4 > 1; since the power 4 makes the sum > 1, then 5 is the 4th greedy power of (Pi/6).

Crossrefs

Cf. A077468-A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(Pi / 6.): pexp := p: a := []: for i from 1 to 800 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;
  • Mathematica
    g[1] = 1; g[n_] := g[n] = With[{x = Pi/6}, Log[x, x^FractionalPart[g[n-1]] - x]]; a[n_] := Sum[Floor[g[k]], {k, 1, n}]; Table[a[n], {n, 1, 64}] (* Jean-François Alcover, Jul 08 2017 *)

Formula

a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(pi/6) and frac(y) = y - floor(y).

Extensions

Corrected by T. D. Noe, Nov 02 2006

A076799 Greedy powers of (e/3): Sum_{n>=1} (e/3)^a(n) = 1.

Original entry on oeis.org

1, 24, 92, 140, 171, 199, 226, 251, 277, 320, 363, 391, 425, 449, 474, 500, 524, 548, 575, 632, 673, 777, 801, 836, 861, 903, 932, 959, 983, 1011, 1054, 1087, 1113, 1148, 1176, 1228, 1261, 1286, 1316, 1348, 1394, 1427, 1452, 1480, 1510, 1536, 1571, 1600
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series Sum_{k=1..n} x^a(k) to exceed unity.

Examples

			a(3)=92 since (e/3) + (e/3)^24 + (e/3)^92 < 1 and (e/3) +(e/3)^24 + (e/3)^91 > 1; since the power 91 makes the sum > 1, then 92 is the 4th greedy power of (e/3).

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 1100: summe := 0.0: p := evalf(exp(1)/3.): pexp := p: a := []: for i from 1 to 3000 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;
  • PARI
    default(realprecision,99); s=1; Le3=1-log(3); for(i=1,50, print1(a=if(i>1,log(s)\Le3,1)","); s-=exp(a*Le3)) \\ M. F. Hasler, Sep 28 2009

Formula

a(n) = Sum_{k=1..n} floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(e/3) and frac(y) = y - floor(y).

Extensions

Some terms corrected (replaced 67,3 with 673 and 153,6 with 1536) by M. F. Hasler, Sep 28 2009

A076800 Greedy powers of (e/4): sum_{n=1..inf} (e/4)^a(n) = 1.

Original entry on oeis.org

1, 3, 13, 38, 41, 54, 57, 60, 63, 67, 73, 75, 88, 91, 95, 98, 101, 109, 116, 122, 125, 129, 131, 142, 145, 151, 159, 163, 169, 172, 176, 190, 200, 205, 210, 215, 217, 228, 235, 241, 250, 252, 266, 271, 276, 280, 283, 296, 298, 311, 315, 318, 323, 326, 329, 334
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series sum_{k=1..n} x^a(k) to exceed unity.

Examples

			a(4)=13 since (e/4) +(e/4)^3 +(e/4)^13 < 1 and (e/4) +(e/4)^3 +(e/4)^12 > 1; since the power 12 makes the sum > 1, then 13 is the 3rd greedy power of (e/4).

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(exp(1)/4.): pexp := p: a := []: for i from 1 to 800 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;

Formula

a(n)=sum_{k=1..n}floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(e/4) and frac(y) = y - floor(y).

Extensions

Corrected by T. D. Noe, Nov 02 2006

A076801 Greedy powers of (e/5): sum_{n=1..inf} (e/5)^a(n) = 1.

Original entry on oeis.org

1, 2, 3, 16, 17, 20, 22, 24, 26, 29, 31, 32, 34, 38, 40, 43, 44, 46, 48, 50, 52, 53, 57, 58, 60, 61, 64, 66, 67, 69, 70, 75, 76, 80, 83, 85, 87, 90, 91, 93, 95, 101, 102, 106, 107, 110, 118, 126, 129, 130, 134, 135, 138, 142, 143, 145, 146, 149, 151, 154, 156, 161
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

Comments

The n-th greedy power of x, when 0.5 < x < 1, is the smallest integer exponent a(n) that does not cause the power series sum_{k=1..n} x^a(k) to exceed unity.

Examples

			a(4)=16 since (e/5) +(e/5)^2 +(e/5)^3 + (e/5)^16 < 1 and (e/5) +(e/5)^2 +(e/5)^3 +(e/5)^15 > 1; since the power 15 makes the sum > 1, then 16 is the 4th greedy power of (e/5).

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(exp(1)/5.): pexp := p: a := []: for i from 1 to 800 do: if summe + pexp < 1 then a := [op(a),i]: summe := summe + pexp: fi: pexp := pexp * p: od: a;

Formula

a(n)=sum_{k=1..n}floor(g_k) where g_1=1, g_{n+1}=log_x(x^frac(g_n) - x) (n>0) at x=(e/5) and frac(y) = y - floor(y).

A360649 The exponents that occur in the greedy representation of 1/2 as a sum of powers of 2/3.

Original entry on oeis.org

2, 8, 11, 14, 16, 26, 33, 38, 45, 48, 51, 53, 65, 69, 72, 80, 83, 89, 94, 101, 105, 109, 115, 118, 123, 131, 139, 142, 148, 152, 157, 160, 164, 170, 176, 179, 182, 185, 188, 193, 197, 208, 214, 220, 223, 225, 232, 234, 240, 243, 247, 250, 254, 258, 261, 271
Offset: 1

Views

Author

James Propp, Feb 15 2023

Keywords

Comments

These numbers correspond to the 1's in the (3/2)-expansion of 1/2, as defined by Renyi.

Examples

			The first power of 2/3 that is smaller than 1/2 is (2/3)^2, so the first term of the sequence is 2. Subtracting (2/3)^2 from 1/2 leaves 1/18. The first power of 2/3 that is less than 1/18 is (2/3)^8, so the next term of the sequence is 8.

Links

Crossrefs

Cf. A058840, A077468.

Programs

  • Maple
    x:= 1/2:
for i from 1 to 100 do
  A[i]:= ceil(log[2/3](x));
  x:= x-(2/3)^A[i];
od:
seq(A[i],i=1..100); # Robert Israel, Feb 15 2023
  • Mathematica
    PositionIndex[RealDigits[1/2, 3/2, 100, -1][[1]]][[2]]

Formula

a(n) = A077468(n+1) - 1. - Andrey Zabolotskiy, Nov 03 2024
Previous Showing 21-26 of 26 results.

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