A353299 - OEIS

%I #9 Apr 10 2022 09:47:27

%S 2,3,2,5,9,10,11,16,13,20,27,27,31,43,37,41,43,47,50,58,53,57,65,83,

%T 69,62,80,84,88,93,88,110,119,117,104,111,116,126,114,140,130,164,166,

%U 132,158,154,166,168,178,178,146,176,192,188,190,203,213,191,224,236,234,238,236,236,251

%N a(n) is the length of the continued fraction for the sum of the reciprocals of the first n primes.

%H Ilya Gutkovskiy, <a href="/A353299/a353299.jpg">Scatterplot of a(n)/(n*log(n)) up to n=10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFraction.html">Continued Fraction</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicSeriesofPrimes.html">Harmonic Series of Primes</a>

%e Sum_{k=1..2} 1/prime(k) = 1/2 + 1/3 = 5/6 = 0 + 1/(1 + 1/5), so a(2) = 3.

%e Sum_{k=1..4} 1/prime(k) = 1/2 + 1/3 + 1/5 + 1/7 = 247/210 = 1 + 1/(5 + 1/(1 + 1/(2 + 1/12))), so a(4) = 5.

%t Table[Length[ContinuedFraction[Sum[1/Prime[k], {k, 1, n}]]], {n, 1, 65}]

%o (PARI) a(n) = #contfrac(sum(k=1, n, 1/prime(k))); \\ _Michel Marcus_, Apr 10 2022

%Y Row lengths of A260615.

%Y Cf. A002110, A024451, A055573.

%K nonn

%O 1,1

%A _Ilya Gutkovskiy_, Apr 09 2022