A179332 a(1)=1; for each n > 1, a(n) is the smallest number such that Sum_{i=1..n} 1/a(i)^2 < sqrt(2).

1, 2, 3, 5, 9, 37, 195, 8584, 1281621, 1325419784, 40182098746967, 203448501599750774078, 4275655952199444141114482835180, 10920781877316031992615629928696178128586477545

Offset: 1

Author

Ben Paul Thurston, Jul 10 2010

Keywords

• easy
• nonn

Comments

In other words, the sequence is the lexicographically first infinite sequence of positive integers whose squared reciprocals sum to less than sqrt(2). After a(1)=1, each term is the smallest number that will not cause the sum of the squares of the reciprocals to exceed the square root of 2.

Examples

			a(1)=1; 1/1^2 = 1;
a(2)=2; 1 + 1/2^2 = 5/4 = 1.25;
a(3)=3; 5/4 + 1/3^2 = 49/36 = 1.3611111111...;
a(4)=5; 49/36 + 1/5^2 = 1261/900 = 1.4011111111...;
a(5)=9; 1261/900 + 1/9^2 = 11449/8100 = 1.4134567901...;
(sums approach sqrt(2) = 1.4142135623...).
		

Crossrefs

Cf. A216245.

Programs

Maple

Digits := 200 : A179332 := proc(n) option remember; if n = 1 then 1; else sqrt(2)-add( 1/procname(i)^2,i=1..n-1) ; ceil( 1/sqrt(%)) ; end if; end proc: seq(A179332(n),n=1..14) ; # R. J. Mathar, Jul 11 2010

Formula

a(n+1) = ceiling(1/sqrt(sqrt(2) - Sum_{i=1..n} 1/a(i)^2)). - R. J. Mathar, Jul 11 2010

Extensions

More terms from R. J. Mathar, Jul 11 2010

Name changed, comments expanded, and example corrected and expanded by Jon E. Schoenfield, Feb 28 2014