A322557 Smallest k such that floor(N*sqrt(Sum_{m=1..k} 6/m^2)) = floor(N*Pi), where N = 10^n.

7, 23, 600, 1611, 10307, 359863, 1461054, 17819245, 266012440, 1619092245, 10634761313, 97509078554, 1203836807622, 10241799698090, 294871290395291, 4004525174270251, 24827457879988026, 112840588371964574, 2064072875704476882, 15243903003939891921

Offset: 0

Author

Zachary Russ, Aug 28 2019

Keywords

• nonn
• base

Comments

6*A007406(k)/A007407(k) = Sum_{m=1..k} 6/m^2.

It seems nearly certain that, for all n >= 0, a(n) = ceiling(z - 1/2 - 1/(12*z)) where z = 6/(Pi^2 - (floor(Pi*10^n)/10^n)^2). - Jon E. Schoenfield, Aug 31 2019

Examples

			floor((10^0)*sqrt(Sum_{m=1..7} 6/m^2)) = 3.
floor((10^1)*sqrt(Sum_{m=1..23} 6/m^2)) = 31.
floor((10^2)*sqrt(Sum_{m=1..600} 6/m^2)) = 314.
floor((10^3)*sqrt(Sum_{m=1..1611} 6/m^2)) = 3141.
floor((10^4)*sqrt(Sum_{m=1..10307} 6/m^2)) = 31415.
floor((10^5)*sqrt(Sum_{m=1..359863} 6/m^2)) = 314159.
		

Links

• Zachary Russ, Klarice Sequence
• Jon E. Schoenfield, Magma program

Crossrefs

Cf. A000796, A013661, A013679, A002388, A274982, A007406, A007407.

Cf. A011545 (floor(Pi*10^n)).

Programs

PARI

a(n) = {my(k = 1); t = floor(10^(n)*Pi); while(floor(10^(n)*sqrt(sum(m = 1, k, 6/m^2))) != t, k++); k; } \\ Jinyuan Wang, Aug 30 2019

Extensions

a(6)-a(19) from Jon E. Schoenfield, Aug 31 2019