This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A105548 #27 Feb 16 2025 08:32:57
%S A105548 2,9,1,1,1,7,3,5,4,1,1,1,3,2,2,1,1,1,1,15,1,3,1,41,6,1,3,1,3,10,1,1,1,
%T A105548 9,9,1,25,1,3,1,1,2,2,2,1,34,59,2,2,2,1,2,2,3,3,1,5,2,21,3,4,10,1,3,
%U A105548 20,2,3,2,1,4,7,1,6,1,6,3,4,1,3,5,6,1,1,4,1,3,6,25,7,2,1,1,2,1,6,1,1,7,1,3,2
%N A105548 Continued fraction expansion of prime nested radical A105546.
%C A105548 Records are: 9,15,41,59,117,153,599,1663,8212,..., . _Robert G. Wilson v_: "It would appear superficially that this constant is normal." Sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + ... = ~ 1.75793275661800... "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; A072449] We know the asymptotic limit of primes and hence that the Prime Nested Radical converges.
%C A105548 Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - _Jonathan Sondow_, Mar 25 2014
%D A105548 Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
%H A105548 Jonathan M. Borwein and G. de Barra, <a href="http://www.jstor.org/stable/2324426">Nested Radicals</a>, Amer. Math. Monthly 98, 735-739, 1991.
%H A105548 J. Sondow and P. Hadjicostas, <a href="http://dx.doi.org/10.1016/j.jmaa.2006.09.081">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl., 332 (2007), 292-314; see pp. 305-306.
%H A105548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NestedRadicalConstant.html">Nested Radical Constant.</a>
%H A105548 Wikipedia, <a href="http://en.wikipedia.org/wiki/Tirukkannapuram_Vijayaraghavan">Tirukkannapuram Vijayaraghavan</a>
%F A105548 Continued fraction expansion of sqrt(2 + sqrt(3 + sqrt(5 + sqrt(7 + sqrt(11 + ... + sqrt(prime(n))))).
%t A105548 f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + k]; k-- ]; s]; ContinuedFraction[ f[100], 101]; (* _Robert G. Wilson v_ *)
%Y A105548 Cf. A000040, A072449, A105546, A239349.
%K A105548 cofr,easy,nonn
%O A105548 0,1
%A A105548 _Jonathan Vos Post_, Apr 14 2005
%E A105548 Offset changed by _Andrew Howroyd_, Aug 03 2024