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A084975 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 A084975 #18 Feb 16 2025 08:32:50
%S A084975 11,127,1361,1693,2503,2999,3299,4327,4861,5623,31469,34123,43391,
%T A084975 44351,58889,156007,370373,492227,604171,1357333,1562051,2010881,
%U A084975 2127269,2238931,4652507,6034393,7230479,8421403,8917663,11114087,20831533
%N A084975 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 A084975 a(n) are the primes p(k+1) 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 A084975 R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
%D A084975 P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.
%H A084975 H. J. Smith, <a href="/A084975/b084975.txt">Table of n, a(n) for n=1..128</a>
%H A084975 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AndricasConjecture.html">Andrica's Conjecture</a>
%H A084975 H. J. Smith, <a href="http://harry-j-smith-memorial.com/PrimeSR/">Andrica's Conjecture</a>
%e A084975 a(3)=1361 because p(218)=1361, p(217)=1327 and Af(217) = sqrt(1361) - sqrt(1327) = 0.463722... is larger than any value of Af(m) for m>217.
%Y A084975 Cf. A078693, A079098, A079296, A084974, A084976, A084977.
%K A084975 nonn
%O A084975 1,1
%A A084975 _Harry J. Smith_, Jun 16 2003