A353299 a(n) is the length of the continued fraction for the sum of the reciprocals of the first n primes.

2, 3, 2, 5, 9, 10, 11, 16, 13, 20, 27, 27, 31, 43, 37, 41, 43, 47, 50, 58, 53, 57, 65, 83, 69, 62, 80, 84, 88, 93, 88, 110, 119, 117, 104, 111, 116, 126, 114, 140, 130, 164, 166, 132, 158, 154, 166, 168, 178, 178, 146, 176, 192, 188, 190, 203, 213, 191, 224, 236, 234, 238, 236, 236, 251

Offset: 1

Author

Ilya Gutkovskiy, Apr 09 2022

Keywords

• nonn

Examples

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

Links

• Ilya Gutkovskiy, Scatterplot of a(n)/(n*log(n)) up to n=10000
• Eric Weisstein's World of Mathematics, Continued Fraction
• Eric Weisstein's World of Mathematics, Harmonic Series of Primes

Crossrefs

Row lengths of A260615.

Cf. A002110, A024451, A055573.

Programs

Mathematica

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

PARI

a(n) = #contfrac(sum(k=1, n, 1/prime(k))); \\ Michel Marcus, Apr 10 2022