A252477 -id:A252477 - cp's OEIS frontend

A074976 a(n) = round(1/(sqrt(prime(n+1)) - sqrt(prime(n)))).

3, 2, 2, 1, 3, 2, 4, 2, 2, 5, 2, 3, 6, 3, 2, 2, 8, 3, 4, 8, 3, 4, 3, 2, 5, 10, 5, 10, 5, 2, 6, 4, 12, 2, 12, 4, 4, 6, 4, 4, 13, 3, 14, 7, 14, 2, 2, 7, 15, 8, 5, 15, 3, 5, 5, 5, 16, 6, 8, 17, 3, 2, 9, 18, 9, 3, 6, 4, 19, 9, 6, 5, 6, 6, 10, 7, 5, 10, 5, 4, 20, 4, 21, 7, 10, 7, 5, 11, 21, 11, 4, 5, 11, 6, 11

Offset: 1

Werner D. Sand and Benoit Cloitre, Oct 06 2002

nonn

If Andrica's conjecture is true, each term in the sequence is >=1.

Since Andrica's conjecture is bounded below only by zero, a(n) is not bounded above.

			a(1) = round(1/(sqrt(3) - sqrt(2))) = round(1/(1.7320.. - 1.4142..)) = round(1/0.3178..) = round(3.1462..) = 3.

C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 482.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Andrica's conjecture

Cf. A000040, A252477.

a074976 n = a074976_list !! (n-1)
a074976_list = map (round . recip) $ zipWith (-) (tail rs) rs
               where rs = map (sqrt . fromIntegral) a000040_list
-- Reinhard Zumkeller, Jan 04 2015

Round[1/(Sqrt[#[[2]]]-Sqrt[#[[1]]])]&/@Partition[Prime[Range[100]],2,1] (* Harvey P. Dale, May 30 2022 *)

a(n) = round(1/(sqrt(prime(n+1))-sqrt(prime(n)))) \\ Michel Marcus, May 22 2013

Conjecture: for n>=4, a(n)>=2. More generally, for any m >=1, the set of k such that a(k)=m is finite. I.e., if n>=217, a(n)>=3; if n>=263, a(n)>=4; if n>=590, a(n)>=5; if n>=3385, a(n)>=6; ...

A079636 Smallest number whose reciprocal fits in the square-root gap of consecutive primes.

Original entry on oeis.org

4, 2, 3, 2, 4, 2, 5, 3, 2, 6, 2, 4, 7, 4, 3, 3, 8, 3, 5, 9, 3, 5, 4, 3, 5, 11, 6, 11, 6, 2, 6, 4, 12, 3, 13, 5, 5, 7, 5, 5, 14, 3, 14, 7, 15, 3, 3, 8, 16, 8, 6, 16, 4, 6, 6, 6, 17, 6, 9, 17, 4, 3, 9, 18, 9, 3, 7, 4, 19, 10, 7, 5, 7, 7, 10, 7, 5, 10, 6, 5, 21

Offset: 1

Rainer Rosenthal, Jan 30 2003

Is the limit of sqrt(P_(n+1)) - sqrt(P_n) = 0?

			a(3) = 3 because p(3)=5, p(4)=7, w=sqrt(5), w'=sqrt(7) and 1/(w'-w)=2.44.

Jim Ferry, sci.math, Jan 30 2003

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000040, A252477.

a:= n-> ((w, v)-> ceil(1/(w-v)))(map(sqrt@ithprime, [n+1, n])[]):
seq(a(n), n=1..81);  # Alois P. Heinz, Aug 23 2025

Ceiling[1/Subtract @@@ Reverse[Partition[Sqrt[Prime[Range[100]]], 2, 1], 2]] (* Paolo Xausa, Aug 24 2025 *)

a(n) = ceiling(1/(w'-w)) where w=sqrt(p(n)) and w'=sqrt(p(n+1)).

a(n) = A252477(n) + 1. - Hugo Pfoertner, Aug 23 2025

More terms from Sean A. Irvine, Aug 23 2025

cp's OEIS Frontend

A074976 a(n) = round(1/(sqrt(prime(n+1)) - sqrt(prime(n)))).

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