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.

Showing 1-10 of 26 results. Next

A077475 Greedy powers of (8/13): Sum_{n>=1} (8/13)^a(n) = 1.

Original entry on oeis.org

1, 2, 11, 14, 25, 28, 30, 37, 39, 41, 43, 46, 48, 51, 54, 57, 60, 64, 66, 71, 76, 78, 80, 82, 84, 90, 95, 101, 103, 106, 110, 113, 115, 117, 127, 133, 135, 140, 146, 152, 157, 160, 162, 165, 167, 170, 173, 179, 181, 185, 189, 196, 200, 203, 206, 209, 212, 215, 220
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.8170308430..., where x=8/13 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002
By the time you reach Sum_{n=1..59} (8/13)^a(n), the difference between that sum and 1 is only 1.6*10^-47.

Examples

			a(3)=11 since (8/13) +(8/13)^2 +(8/13)^11 < 1 and (8/13)+(8/13)^2+(8/13)^10 >1.

Links

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077474.

Programs

  • Maple
    V:= Vector(100):
V[1]:= 1: T:= 1 - 8/13:
for n from 2 to 100 do
V[n]:= -floor(log[13/8](T));
T:= T - (8/13)^V[n];
od:
convert(V,list); # Robert Israel, Aug 11 2020
  • Mathematica
    s = 0; a = {}; Do[ If[s + (8/13)^n < 1, s = s + (8/13)^n; a = Append[a, n]], {n, 1, 250}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[8/13], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=8/13 and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.7... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002.
Extended by Benoit Cloitre, Nov 06 2002

A077471 Greedy powers of (4/7): Sum_{n>=1} (4/7)^a(n) = 1.

Original entry on oeis.org

1, 2, 5, 6, 10, 11, 14, 18, 19, 23, 27, 29, 30, 35, 36, 39, 55, 56, 60, 62, 64, 73, 75, 78, 79, 83, 84, 87, 95, 99, 104, 111, 113, 121, 122, 126, 133, 134, 141, 143, 147, 151, 152, 161, 162, 165, 169, 171, 173, 175, 176, 179, 182, 183, 186, 189, 197, 202, 205, 207
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.0486255758..., where x=4/7 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=5 since (4/7) +(4/7)^2 +(4/7)^5 < 1 and (4/7) +(4/7)^2 +(4/7)^4 > 1.

Links

Crossrefs

Cf. A077468, A077469, A077470, A077472, A077473, A077474, A077475.

Programs

  • Maple
    s:= 0: count:= 0:
R:= NULL;
for n from 1 while count < 100 do
  t:= (4/7)^n;
  if s+t < 1 then count:= count+1; R:= R, n; s:= s+t fi
od:
R; # Robert Israel, Jun 01 2018
  • Mathematica
    s = 0; a = {}; Do[ If[s + (4/7)^n < 1, s = s + (4/7)^n; a = Append[a, n]], {n, 1, 208}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[4/7], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g_n) - x) at x=4/7 and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.3... - Benoit Cloitre

Extensions

Extended by Benoit Cloitre, Nov 06 2002
Edited and extended by Robert G. Wilson v, Nov 08 2002

A077469 Greedy powers of (3/4): Sum_{n>=1} (3/4)^a(n) = 1.

Original entry on oeis.org

1, 5, 16, 21, 29, 35, 39, 52, 57, 63, 68, 76, 82, 88, 93, 99, 106, 113, 118, 127, 134, 150, 155, 160, 167, 172, 182, 192, 197, 209, 215, 224, 229, 237, 242, 246, 260, 265, 272, 278, 289, 293, 310, 315, 320, 330, 337, 346, 353, 373, 379, 384, 390, 396, 405, 416
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 8.0547475948..., where x=3/4 and m=floor(log(1-x)/log(x))=4. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=9 since (3/4) +(3/4)^5 +(3/4)^16 < 1 and (3/4) +(3/4)^5 +(3/4)^15 > 1.

Crossrefs

Cf. A077468, A077470, A077471, A077472, A077473, A077474, A077475.

Programs

  • Mathematica
    s = 0; a = {}; Do[ If[s + (3/4)^n < 1, s = s + (3/4)^n; a = Append[a, n]], {n, 1, 428}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[3/4], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1)=log_x(x^frac(g_n) - x) at x=(3/4) and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 8.0... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002

A077470 Greedy powers of (3/5): Sum_{n>=1} (3/5)^a(n) = 1.

Original entry on oeis.org

1, 2, 7, 9, 13, 15, 18, 20, 22, 27, 31, 37, 39, 40, 49, 55, 57, 66, 68, 70, 71, 77, 79, 81, 82, 87, 94, 98, 104, 106, 107, 114, 117, 120, 121, 129, 133, 136, 138, 141, 150, 151, 157, 158, 163, 166, 169, 173, 181, 184, 192, 198, 199, 205, 207, 209, 213, 218, 224
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.5151141759..., where x=3/5 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=7 since (3/5) +(3/5)^2 +(3/5)^7 < 1 and (3/5) +(3/5)^2 +(3/5)^6 > 1.

Crossrefs

Cf. A077468, A077469, A077471, A077472, A077473, A077474, A077475.

Programs

  • Mathematica
    s = 0; a = {}; Do[ If[s + (3/5)^n < 1, s = s + (3/5)^n; a = Append[a, n]], {n, 1, 226}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[3/5], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=3/5 and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.7... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002

A077472 Greedy powers of (5/8): Sum_{n>=1} (5/8)^a(n) = 1.

Original entry on oeis.org

1, 3, 5, 8, 10, 13, 15, 23, 26, 30, 33, 36, 38, 46, 48, 51, 53, 57, 61, 64, 66, 69, 72, 76, 78, 84, 88, 93, 95, 104, 106, 110, 115, 117, 121, 126, 129, 131, 136, 138, 143, 148, 150, 152, 157, 160, 164, 169, 172, 175, 179, 181, 185, 187, 191, 196, 198, 201, 203
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 3.9944918847..., where x=5/8 and m=floor(log(1-x)/log(x))=2. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=5 since (5/8) +(5/8)^3 +(5/8)^5 < 1 and (5/8) +(5/8)^3 +(5/8)^4 > 1.

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077473, A077474, A077475.

Programs

  • Mathematica
    s = 0; a = {}; Do[ If[s + (5/8)^n < 1, s = s + (5/8)^n; a = Append[a, n]], {n, 1, 210}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[5/8], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=(5/8) and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 3.4... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002

A077473 Greedy powers of (5/9): Sum_{n>=1} (5/9)^a(n) = 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 13, 18, 21, 24, 27, 28, 30, 32, 35, 37, 40, 43, 45, 50, 51, 59, 62, 64, 73, 76, 79, 82, 83, 86, 87, 93, 96, 99, 100, 103, 106, 108, 110, 112, 113, 117, 118, 121, 123, 126, 127, 131, 137, 139, 140, 143, 145, 146, 148, 154, 155, 157, 163, 165, 166
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 2.8326013771..., where x=5/9 and m=floor(log(1-x)/log(x))=1. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=4 since (5/9) +(5/9)^2 +(5/9)^4 < 1 and (5/9) +(5/9)^2 +(5/9)^3 > 1.

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077474, A077475.

Programs

  • Mathematica
    s = 0; a = {}; Do[ If[s + (5/9)^n < 1, s = s + (5/9)^n; a = Append[a, n]], {n, 1, 173}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[5/9], 20]

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=5/9 and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 2.8... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002. Also extended by Benoit Cloitre, Nov 06 2002

A077474 Greedy powers of (7/10): Sum_{n>=1} (7/10)^a(n) = 1.

Original entry on oeis.org

1, 4, 8, 18, 21, 28, 31, 36, 41, 44, 55, 58, 71, 76, 79, 84, 88, 108, 125, 135, 141, 148, 155, 158, 164, 175, 180, 185, 195, 198, 218, 225, 230, 237, 242, 246, 250, 254, 259, 263, 268, 276, 281, 300, 305, 310, 317, 321, 326, 329, 334, 340, 343, 351, 359, 364
Offset: 1

Views

Author

Paul D. Hanna, Nov 06 2002

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.
A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 5.9293123466..., where x=7/10 and m=floor(log(1-x)/log(x))=3. - Paul D. Hanna, Nov 16 2002

Examples

			a(3)=8 since (7/10) +(7/10)^3 +(7/10)^8 < 1 and (7/10) +(7/10)^3 +(7/10)^7 > 1.

Crossrefs

Cf. A077468, A077469, A077470, A077471, A077472, A077473, A077475.

Programs

  • Mathematica
    s = 0; a = {}; Do[ If[s + (7/10)^n < 1, s = s + (7/10)^n; a = Append[a, n]], {n, 1, 368}]; a
heuristiclimit[x_] := (m=Floor[Log[x, 1-x]])+1/24+Log[x, Product[1+x^n, {n, 1, m-1}]/DedekindEta[I Log[x]/-Pi]*DedekindEta[ -I Log[x]/2/Pi]]; N[heuristiclimit[7/10], 20]

Formula

a(n) = Sum_{k=1..n} floor(g(k)) where g(1)=1, g(n+1) = log_x(x^frac(g(n)) - x) at x=7/10 and frac(y) = y - floor(y).
a(n) seems to be asymptotic to c*n with c around 6... - Benoit Cloitre

Extensions

Edited and extended by Robert G. Wilson v, Nov 08 2002; also extended by Benoit Cloitre, Nov 06 2002

A076802 Greedy powers of the gamma constant (0.577215664...) Sum_{n=1..infinity} (gamma)^a(n) = 1.

Original entry on oeis.org

1, 2, 5, 7, 10, 18, 20, 22, 23, 26, 30, 33, 37, 41, 44, 46, 48, 49, 53, 56, 58, 59, 68, 69, 75, 77, 78, 81, 88, 90, 94, 96, 98, 100, 102, 105, 106, 109, 111, 116, 120, 122, 124, 126, 132, 135, 137, 140, 145, 152, 155, 157, 158, 162, 165, 168, 171, 174, 176, 178
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)=5 since (gamma) + (gamma)^2 + (gamma)^5 < 1 and (gamma) + (gamma)^2 + (gamma)^4 > 1; the power 4 makes the sum > 1, so 5 is the 3rd greedy power of gamma.

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(gamma): 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 = gamma and frac(y) = y - floor(y).

Extensions

Corrected by T. D. Noe, Nov 02 2006

A076796 Greedy powers of Pi/4: Sum_{n>=1} (Pi/4)^a(n) = 1.

Original entry on oeis.org

1, 7, 15, 24, 32, 39, 47, 59, 79, 88, 102, 111, 134, 148, 158, 164, 172, 190, 206, 214, 220, 233, 241, 251, 263, 271, 283, 292, 307, 314, 322, 329, 337, 350, 358, 364, 373, 384, 399, 413, 438, 446, 456, 462, 475, 481, 494, 502, 516, 529, 536, 552, 559, 567
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/4 + (Pi/4)^7 + (Pi/4)^15 < 1 and Pi/4 + (Pi/4)^7 + (Pi/4)^14 > 1; since the power 14 makes the sum > 1, 15 is the 3rd greedy power of Pi/4, so a(3)=15.

Crossrefs

Cf. A077468 - A077475.

Programs

  • Maple
    Digits := 400: summe := 0.0: p := evalf(Pi / 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=Pi/4 and frac(y) = y - floor(y).

Extensions

Typos in data corrected by Sean A. Irvine, Apr 16 2025

A079930 Greedy powers of (1/sqrt(e)): Sum_{n>=1} (1/sqrt(e))^a(n) = 1.

Original entry on oeis.org

1, 2, 8, 10, 16, 18, 19, 26, 30, 36, 38, 41, 43, 45, 50, 51, 59, 65, 68, 70, 74, 75, 82, 84, 87, 89, 91, 94, 96, 99, 101, 103, 107, 113, 116, 117, 124, 127, 129, 136, 138, 142, 145, 149, 156, 161, 164, 166, 168, 170, 172, 176, 181, 183, 185, 187, 189, 192, 194, 196
Offset: 1

Views

Author

Ulrich Schimke (ulrschimke(AT)aol.com), Jan 16 2003

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)=8 since (1/sqrt(e)) + (1/sqrt(e))^2 + (1/sqrt(e))^8 < 1 and (1/sqrt(e)) + (1/sqrt(e))^2 + (1/sqrt(e))^7 > 1; the power 7 makes the sum > 1, so 8 is the 3rd greedy power of (1/sqrt(e)).

Crossrefs

Cf. A076796-A076802, A077468-A077475, A079931-A079933.

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=(1/sqrt(e)) and frac(y) = y - floor(y).
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