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A084974 Primes 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 A084974 #21 Feb 16 2025 08:32:50
%S A084974 7,113,1327,1669,2477,2971,3271,4297,4831,5591,31397,34061,43331,
%T A084974 44293,58831,155921,370261,492113,604073,1357201,1561919,2010733,
%U A084974 2127163,2238823,4652353,6034247,7230331,8421251,8917523,11113933,20831323
%N A084974 Primes 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 A084974 a(n) are the primes p(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 A084974 R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
%D A084974 P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.
%H A084974 H. J. Smith, <a href="/A084974/b084974.txt">Table of n, a(n) for n = 1..128</a>
%H A084974 H. J. Smith, <a href="https://www.oocities.org/hjsmithh/PrimeSR/index.html">Andrica's Conjecture</a>
%H A084974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndricasConjecture.html">Andrica's Conjecture</a>.
%e A084974 a(3)=1327 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 A084974 Cf. A078693, A079098, A079296, A084975, A084976, A084977.
%K A084974 nonn
%O A084974 1,1
%A A084974 _Harry J. Smith_, Jun 16 2003