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A084976 Values of k that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

%I A084976 #16 Feb 16 2025 08:32:50
%S A084976 4,30,217,263,367,429,462,590,650,738,3385,3644,4522,4612,5949,14357,
%T A084976 31545,40933,49414,104071,118505,149689,157680,165326,325852,415069,
%U A084976 491237,566214,597311,733588,1319945,1736516,2850174,2857960,3183065
%N A084976 Values of k that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.
%C A084976 a(n) are values of k such that Af(k) > Af(m) for all m > k. This sequence relies on a heuristic calculation and there is no proof that it is correct.
%D A084976 R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
%D A084976 P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.
%H A084976 H. J. Smith, <a href="/A084976/b084976.txt">Table of n, a(n) for n=1..128</a>
%H A084976 H. J. Smith, <a href="http://harry-j-smith-memorial.com/PrimeSR/">Andrica's Conjecture</a>
%H A084976 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndricasConjecture.html">Andrica's conjecture.</a>
%e A084976 a(3)=217 because p(217)=1327, p(218)=1361 and Af(217) =sqrt(1361) - sqrt(1327) = 0.463722... is larger than any value of Af(m)for m>217.
%Y A084976 Cf. A078693, A079098, A079296, A084974, A084975, A084977.
%K A084976 nonn
%O A084976 1,1
%A A084976 _Harry J. Smith_, Jun 16 2003