A077468 -id:A077468 - cp's OEIS frontend

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

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

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

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(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)).

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

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).

A073536 Breaking indices for A058842 (i.e., n such that A058842(n) is not equal to 3*A058842 (n-1) ).

Original entry on oeis.org

3, 9, 12, 15, 17, 27, 34, 39, 46, 49, 52, 54, 66, 70, 73, 81, 84, 90, 95, 102, 106, 110, 116, 119, 124, 132, 140, 143, 149, 153, 158, 161, 165, 171, 177, 180, 183, 186, 189, 194, 198, 209, 215, 221, 224, 226, 233, 235, 241, 244, 248, 251, 255, 259, 262, 272

Offset: 1

Benoit Cloitre, Aug 27 2002

Cf. A058842.

It seems that a(n) = 5*n +O(log(n)) and that a(n)=5*n for infinitely many values of n.

It appears that a(n)=A077468(n+1). - Benoit Cloitre, Jun 04 2004

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

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

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(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)).

Cf. A076796-A076802, A077468-A077475, A079930, A079932, A079933.

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

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

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(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)).

Cf. A076796-A076802, A077468-A077475, A079930, A079931, A079933.

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).

A080056 Greedy powers of (2/Pi): Sum_{n=1..inf} (2/Pi)^a(n) = 1.

Original entry on oeis.org

1, 3, 5, 16, 22, 24, 28, 34, 37, 43, 45, 49, 51, 54, 57, 59, 65, 68, 70, 74, 80, 88, 94, 97, 100, 103, 108, 111, 113, 116, 122, 127, 129, 132, 137, 141, 143, 148, 151, 156, 161, 164, 166, 172, 174, 177, 184, 189, 202, 204, 208, 213, 216, 219, 225, 227, 238, 247

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

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..inf} log(1 + x^n)/log(x) = 4.2164448079..., where x=(2/Pi) and m=floor(log(1-x)/log(x))=2.

See A077468 for Mathematica program by Robert G. Wilson v.

			a(3)=5 since (2/Pi) +(2/Pi)^3 +(2/Pi)^5 < 1 and (2/Pi) +(2/Pi)^3 +(2/Pi)^k > 1 for 3<k<5.

Cf. A077468, A080055, A080057.

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=(2/Pi) 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

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

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.

			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.

Cf. A077468, A076802, A080056, A080058.

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).

A080059 Greedy powers of (1/zeta(3)): Sum_{n>=1} (1/zeta(3))^a(n) = 1, where 1/zeta(3) = .83190737258070746868...

Original entry on oeis.org

1, 10, 26, 38, 54, 64, 80, 98, 115, 126, 136, 147, 158, 171, 181, 196, 206, 226, 243, 257, 267, 279, 293, 306, 324, 334, 355, 365, 378, 388, 398, 410, 432, 442, 455, 468, 491, 501, 519, 534, 545, 560, 572, 582, 593, 610, 628, 638, 650, 663, 672, 691, 704, 715

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

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) = 14.874449248373..., where x=(1/zeta(3)) and m=floor(log(1-x)/log(x))=9.

See A077468 for Mathematica program by Robert G. Wilson v.

			a(3)=26 since (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^26 < 1 and (1/zeta(3)) +(1/zeta(3))^10 +(1/zeta(3))^k > 1 for 10<k<26.

Cf. A077468, A080059.

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

A079450 a(n) = 2^(n-1)u(n) where u(1)=1 and u(n) = frac(3/2u(n-1)) + 1.

Original entry on oeis.org

1, 3, 5, 15, 29, 55, 101, 175, 269, 807, 1397, 2143, 6429, 11095, 16901, 50703, 86573, 259719, 517013, 1026751, 2031677, 3997879, 7799333, 15009391, 28250957, 51198439, 86486453, 259459359, 509942621, 992956951, 1905129029, 3567903439

Offset: 1

Benoit Cloitre, Jan 13 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

u[1]:= 1; u[n_]:= u[n]= FractionalPart[3*u[n-1]/2] +1; a[n_]:= 2^(n-1)* u[n]; Table[a[n], {n, 1, 40}] (* G. C. Greubel, Jan 18 2019 *)

a(n+1)<=3*a(n). Conjecture : a(n+1)=3*a(n) if and only if n is a greedy power of (2/3) (i.e. n is in A077468)

A080055 Greedy powers of log(2): Sum_{n>=1} (log(2))^a(n) = 1.

Original entry on oeis.org

1, 4, 8, 11, 15, 20, 23, 30, 38, 43, 49, 54, 60, 65, 72, 78, 85, 90, 93, 100, 104, 108, 111, 115, 118, 122, 128, 132, 140, 144, 147, 152, 156, 159, 171, 174, 178, 181, 188, 191, 196, 203, 206, 210, 213, 232, 244, 248, 256, 260, 265, 269, 272, 276, 285, 289, 293

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

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) = 5.7114827587..., where x = log(2) and m = floor(log(1-x)/log(x))=3.

See A077468 for Mathematica program by Robert G. Wilson v.

			a(3)=8 since (log(2)) + (log(2))^4 + (log(2))^8 < 1 and (log(2)) + (log(2))^4 + (log(2))^k > 1 for 4 < k < 8.

Cf. A077468, A080056.

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=log(2) and frac(y) = y - floor(y).

A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

Original entry on oeis.org

1, 2, 8, 12, 14, 16, 25, 39, 42, 44, 46, 49, 51, 53, 59, 70, 73, 78, 81, 83, 85, 86, 101, 103, 105, 116, 118, 119, 126, 130, 135, 137, 139, 142, 144, 147, 148, 158, 161, 163, 170, 171, 178, 181, 186, 188, 190, 192, 194, 195, 204, 207, 209, 212, 216, 219, 224, 229

Offset: 1

Benoit Cloitre and Paul D. Hanna, Jan 23 2003

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) = 3.66565771136..., where x=(1/zeta(2)) and m=floor(log(1-x)/log(x))=1.

See A077468 for Mathematica program by Robert G. Wilson v.

			a(3)=8 since (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^8 < 1 and (1/zeta(2)) +(1/zeta(2))^2 +(1/zeta(2))^k > 1 for 2<k<8.

Cf. A077468, A080057, A080059, A229099.

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/zeta(2)) and frac(y) = y - floor(y).

cp's OEIS Frontend

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

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A073536 Breaking indices for A058842 (i.e., n such that A058842(n) is not equal to 3*A058842 (n-1) ).

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

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

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A080056 Greedy powers of (2/Pi): Sum_{n=1..inf} (2/Pi)^a(n) = 1.

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

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A080059 Greedy powers of (1/zeta(3)): Sum_{n>=1} (1/zeta(3))^a(n) = 1, where 1/zeta(3) = .83190737258070746868...

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A079450 a(n) = 2^(n-1)*u(n) where u(1)=1 and u(n) = frac(3/2*u(n-1)) + 1.

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A080055 Greedy powers of log(2): Sum_{n>=1} (log(2))^a(n) = 1.

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A080058 Greedy powers of (1/zeta(2)): Sum_{n>=1} (1/zeta(2))^a(n) = 1, where 1/zeta(2) = 6/Pi^2 = .607927101854...

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A079450 a(n) = 2^(n-1)u(n) where u(1)=1 and u(n) = frac(3/2u(n-1)) + 1.