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 1-5 of 5 results.

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).

A079933 Greedy powers of (1/sqrt(3)): sum_{n=1..inf} (1/sqrt(3))^a(n) = 1.

Original entry on oeis.org

1, 2, 5, 7, 11, 12, 19, 22, 27, 33, 37, 39, 42, 44, 53, 54, 60, 62, 68, 69, 75, 77, 78, 83, 86, 87, 91, 94, 97, 100, 101, 105, 106, 110, 113, 115, 116, 120, 121, 125, 129, 131, 132, 137, 141, 144, 148, 149, 152, 155, 157, 166, 171, 173, 178, 179, 184, 186, 189, 191
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)=5 since (1/sqrt(3)) + (1/sqrt(3))^2 + (1/sqrt(3))^5 < 1 and (1/sqrt(3)) + (1/sqrt(3))^2 + (1/sqrt(3))^4 > 1; the power 4 makes the sum > 1, so 5 is the 3rd greedy power of (1/sqrt(3)).

Crossrefs

Cf. A076796-A076802, A077468-A077475, A079930-A079932.

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(3)) and frac(y) = y - floor(y).

A079931 Greedy powers of (1/sqrt(Pi)): Sum_{n>=1} (1/sqrt(Pi))^a(n) = 1.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 20, 22, 23, 32, 33, 36, 39, 42, 43, 46, 47, 50, 51, 55, 59, 60, 63, 69, 74, 77, 80, 82, 87, 92, 94, 97, 100, 102, 105, 107, 111, 113, 114, 117, 119, 122, 126, 128, 129, 134, 141, 142, 146, 147, 150, 151, 154, 157, 160, 162, 165, 167, 168, 171, 175
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)=4 since (1/sqrt(Pi)) + (1/sqrt(Pi))^2 + (1/sqrt(Pi))^4 < 1 and (1/sqrt(Pi)) + (1/sqrt(Pi))^2 + (1/sqrt(Pi))^3 > 1; the power 3 makes the sum > 1, so 4 is the 3rd greedy power of (1/sqrt(Pi)).

Crossrefs

Cf. A076796-A076802, A077468-A077475, A079930, A079932, 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(Pi)) and frac(y) = y - floor(y).

A079932 Greedy powers of (1/sqrt(2)): sum_{n=1..inf} (1/sqrt(2))^a(n) = 1.

Original entry on oeis.org

1, 4, 10, 13, 22, 27, 32, 36, 40, 49, 54, 62, 66, 71, 80, 91, 97, 102, 109, 114, 120, 124, 127, 138, 146, 149, 159, 165, 169, 180, 184, 187, 194, 202, 208, 219, 224, 231, 235, 248, 258, 263, 266, 274, 281, 287, 294, 300, 304, 308, 316, 323, 329, 337, 343, 350
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)=10 since (1/sqrt(2)) + (1/sqrt(2))^4 + (1/sqrt(2))^10 < 1 and (1/sqrt(2)) + (1/sqrt(2))^4 + (1/sqrt(2))^9 > 1; the power 9 makes the sum > 1, so 10 is the 3rd greedy power of (1/sqrt(2)).

Crossrefs

Cf. A076796-A076802, A077468-A077475, A079930, 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(2)) and frac(y) = y - floor(y).

A080057 Greedy powers of exp(-gamma): Sum_{n>=1} exp(-gamma)^a(n) = 1, where exp(-gamma) = exp(-.57721566490153286...) = .561459483566885169...

Original entry on oeis.org

1, 2, 4, 7, 9, 13, 15, 17, 20, 21, 23, 27, 29, 34, 35, 38, 40, 42, 43, 46, 48, 49, 51, 54, 57, 58, 61, 64, 65, 68, 73, 74, 80, 83, 85, 87, 89, 98, 100, 101, 104, 105, 107, 110, 113, 116, 117, 120, 122, 123, 126, 128, 132, 136, 139, 142, 149, 152, 156, 157, 160, 161, 163
Offset: 1

Views

Author

Benoit Cloitre and Paul D. Hanna, Jan 23 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. A heuristic argument suggests that the limit of a(n)/n is m - Sum_{n>=m} log(1 + x^n)/log(x) = 2.909795625992782..., where x=exp(-gamma) and m=floor(log(1-x)/log(x))=1.
See A077468 for Mathematica program by Robert G. Wilson v.

Examples

			a(3)=4 since exp(-gamma) + exp(-gamma)^2 + exp(-gamma)^4 < 1 and exp(-gamma) + exp(-gamma)^2 + exp(-gamma)^k > 1 for 2<k<4.

Crossrefs

Cf. A077468, A076802, A080056, A080058.

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=(exp(-Gamma)) and frac(y) = y - floor(y).
Showing 1-5 of 5 results.

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