A073333 -id:A073333 - cp's OEIS frontend

A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

5, 8, 1, 5, 2, 5, 0, 0, 4, 5, 9, 2, 2, 1, 4, 6, 5, 4, 3, 9, 9, 1, 5, 1, 7, 0, 4, 8, 1, 8, 0, 0, 4, 4, 6, 1, 9, 5, 5, 8, 6, 7, 5, 4, 0, 4, 9, 7, 2, 4, 6, 4, 4, 1, 1, 0, 0, 4, 7, 9, 4, 2, 3, 2, 6, 0, 9, 6, 7, 4, 6, 4, 5, 4, 1, 9, 6, 8, 6, 1, 4, 1, 2, 0, 2, 7, 6, 1, 4, 5, 2, 4, 3, 4, 0, 5, 4, 6, 9, 3

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			0.58152500459221465439915170481800446195586754...

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A000040, A064442, A068996, A073333, A084255, A085825, A152062, A191504, A191815, A233588, A247856.

A336112 a(n) is the least number k such that the Sum_{i=0..k} sqrt(k) equals or exceeds n.

Original entry on oeis.org

0, 1, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23

Offset: 0

Robert G. Wilson v, Jul 08 2020

nonn

Inspired by A045880.
Let c = (9/4)^(1/3) = (3/2)^(2/3) ~ 1.310370697..., then a(n) ~ c*n^(2/3).
a(10^k) for k>= 0: 1, 6, 28, 131, 608, 2823, 13104, 60822, 282311, 1310371, 6082202, 28231081, 131037070, 608220200, ..., .

			a(0) = 0 since the sqrt(0) = 0;
a(1) = 1 since the sqrt(0) + sqrt(1) = 1;
a(2) = 2 since the sqrt(0) + sqrt(1) + sqrt(2) ~ 2.41421... which exceeds 2;
a(3) = 3 since the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which easily exceeds 3;
a(4) = 3 because the sqrt(0) + sqrt(1) + sqrt(2) + sqrt(3) ~ 4.146264... which barely exceeds 4; etc.

Cf. A025224, A045880, A073333.

f[n_] := Block[{k = s = 0}, While[s < n, k++; s = s + Sqrt@k]; k]; Array[f, 75, 0]

a(n) = my(s=0, k=0); while ((s+=sqrt(k)) < n, k++); k; \\ Michel Marcus, Jul 09 2020

a(k*n) ~ k^(2/3)*a(n).

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A330867 Decimal expansion of the continued fraction 1/(1 + 2/(2 + 3/(3 + 5/(5 + 7/(7 + ... + prime(k)/(prime(k) + ...)))))).

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A336112 a(n) is the least number k such that the Sum_{i=0..k} sqrt(k) equals or exceeds n.

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