A079932 -id:A079932

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A079933

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..60.

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).

EXAMPLE

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.

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

A079931

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..61.

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).

EXAMPLE

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.

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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