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

%I #75 Sep 03 2019 16:53:59

%S 7,23,600,1611,10307,359863,1461054,17819245,266012440,1619092245,

%T 10634761313,97509078554,1203836807622,10241799698090,294871290395291,

%U 4004525174270251,24827457879988026,112840588371964574,2064072875704476882,15243903003939891921

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

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

%C 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

%H Zachary Russ, <a href="https://www.desmos.com/calculator/jdnvwymx9p">Klarice Sequence</a>

%H Jon E. Schoenfield, <a href="/A322557/a322557.txt">Magma program</a>

%e floor((10^0)*sqrt(Sum_{m=1..7} 6/m^2)) = 3.

%e floor((10^1)*sqrt(Sum_{m=1..23} 6/m^2)) = 31.

%e floor((10^2)*sqrt(Sum_{m=1..600} 6/m^2)) = 314.

%e floor((10^3)*sqrt(Sum_{m=1..1611} 6/m^2)) = 3141.

%e floor((10^4)*sqrt(Sum_{m=1..10307} 6/m^2)) = 31415.

%e floor((10^5)*sqrt(Sum_{m=1..359863} 6/m^2)) = 314159.

%o (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

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

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

%K nonn,base

%O 0,1

%A _Zachary Russ_, Aug 28 2019

%E a(6)-a(19) from _Jon E. Schoenfield_, Aug 31 2019

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)