A084974 Primes p(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.

7, 113, 1327, 1669, 2477, 2971, 3271, 4297, 4831, 5591, 31397, 34061, 43331, 44293, 58831, 155921, 370261, 492113, 604073, 1357201, 1561919, 2010733, 2127163, 2238823, 4652353, 6034247, 7230331, 8421251, 8917523, 11113933, 20831323

Offset

1,1

Comments

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.

References

R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.

P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.

Links

Harry J. Smith, Table of n, a(n) for n = 1..128

J. M. Flagg, Louis H. Kauffman, and Divyamaan Sahoo, Primes Between Squares - Commentary on Appendix 8 of Laws Of Form, arXiv:2511.05603 [math.NT], 2025. See pp. 24, 52.

Harry J. Smith, Andrica's Conjecture.

Eric Weisstein's World of Mathematics, Andrica's Conjecture.

Formula

a(n) = prevprime(A084975(n)). - Sean A. Irvine, May 23 2026

Example

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.

Crossrefs

Cf. A078693, A079098, A079296, A084975, A084976, A084977.

Sequence in context: A079296 A081531 A142537 * A383591 A156240 A152927

Adjacent sequences: A084971 A084972 A084973 * A084975 A084976 A084977

Keyword

nonn

Author

Harry J. Smith, Jun 16 2003

Status

approved